# Applicable Mathematics/Printable version

Applicable Mathematics

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# Systems of Equations

Home - 1 - 2 - 3 - 4 - 5 - 6- 7 - 8 - 9 - 10

A good deal of real world problems can be represented by various equations. Often, we will have more than one equation for a given problem.

# Substitution

Substitution uses letters, such as x, y, or z, as representations of unknown values. These letters are used in both equations and expressions as tools to solve many different types of problems. In some cases, the value of the letter is known. If so, by using the substitution method, a numerical value replaces the letter given. Then, after the letter is replaced by a number, the expression or equation is simplified.

## Introductory Examples

### People & Feet

In a room of people we know there are twice as many feet as people, and we can represent this with the equation ${\displaystyle f=2p}$, where ${\displaystyle f}$ represents the number of feet, and ${\displaystyle p}$ represents the number of people. Knowing there are 20 people, then we can make another equation ${\displaystyle p=20}$. Listing out the two equations we have:

 ${\displaystyle f}$ ${\displaystyle =2p}$ ${\displaystyle p}$ ${\displaystyle =20}$ Substituting for ${\displaystyle p}$, we get the equation ${\displaystyle f}$ ${\displaystyle =2\times 20}$ ${\displaystyle =40}$

So we know there are 40 feet.

### Houses & Floorspace

Other situations can get more complex though. Suppose that your neighbor's house is 1.5 times as large as yours, but if you don't count your neighbor's 50 square unit basement, they are the same size. How many square units are the respective houses? Let ${\displaystyle y}$ be the area of your entire floor, and ${\displaystyle n}$ the area of your neighbor's floor. The problem can represented by the two equations.

 ${\displaystyle n}$ ${\displaystyle =1.5y}$ ${\displaystyle y}$ ${\displaystyle =n-50}$

Here, you have two equations as before, but this time they both have two variables, while in the last example, one equation had one variable and the other had two. However, you can still use substitution though you just have to substitute with the equation's expression instead of a constant. Now we substitute ${\displaystyle n}$ in the second equation for the right-hand side of the first equation:

 ${\displaystyle 1.5y-50}$ ${\displaystyle =y}$ Subtract y from both sides ${\displaystyle 0.5y-50}$ ${\displaystyle =0}$ Add 50 to both sides ${\displaystyle 0.5y}$ ${\displaystyle =50}$ Multiply both sides by 2 ${\displaystyle y}$ ${\displaystyle =100}$

Now we can substitute the area of your floor into the first equation to get the area of your neighbours floor:

 ${\displaystyle n}$ ${\displaystyle =1.5\times 100}$ ${\displaystyle =150}$

## Systems of Linear Equations

Above we covered two real world examples (albeit simplified) that systems of equations are useful for solving. Equations composed of two or more linear functions are called Linear equations. Sets of these linear equations are called System of linear equations. Simply put, linear equations can only be solved if the number of Variables is equal to or less than the number of equations provided. The most common way of solving systems of equations is to use substitution, as shown above. However, we can represent the equations using matrices, allowing us to see patterns easier and perform operations more easily. Lets start with a set of 3 equations:

 ${\displaystyle x+2y+3z}$ ${\displaystyle =12}$ ${\displaystyle 2x+3y+z}$ ${\displaystyle =17}$ ${\displaystyle 3x+y+2z}$ ${\displaystyle =19}$

The matrix on the left represents the coefficients of the variables, and is called a coefficient matrix. On the right the right-hand side of the equation is included in the matrix, giving us what is called an augmented matrix.

 ${\displaystyle {\begin{bmatrix}1&2&3\\2&3&1\\3&1&2\end{bmatrix}}}$ → ${\displaystyle {\begin{bmatrix}1&2&3&\vdots &12\\2&3&1&\vdots &17\\3&1&2&\vdots &19\end{bmatrix}}}$

Solving the above equation would look like the table on the left below normally, whereas the matrix solution would look like the table on the right.

 ${\displaystyle e_{1}}$ ${\displaystyle x+2y+3z}$ ${\displaystyle =12}$ ${\displaystyle e_{2}}$ ${\displaystyle 2x+3y+1z}$ ${\displaystyle =17}$ ${\displaystyle e_{3}}$ ${\displaystyle 3x+y+2z}$ ${\displaystyle =19}$ ${\displaystyle e_{2}-2e_{1}}$ ${\displaystyle -y-5z}$ ${\displaystyle =-7}$ ${\displaystyle e_{3}-3e_{1}}$ ${\displaystyle -5y-7z}$ ${\displaystyle =-17}$ ${\displaystyle e_{5}-5e_{4}}$ ${\displaystyle 18z}$ ${\displaystyle =18}$ ${\displaystyle e_{6} \over 18}$ ${\displaystyle z}$ ${\displaystyle =1}$ ${\displaystyle e_{4}}$ ${\displaystyle -y-5(1)}$ ${\displaystyle =-7}$ ${\displaystyle -5-e_{4}}$ ${\displaystyle y}$ ${\displaystyle =2}$ ${\displaystyle e_{1}}$ ${\displaystyle x+2(2)+3(1)}$ ${\displaystyle =12}$ ${\displaystyle e_{2}-7}$ ${\displaystyle x}$ ${\displaystyle =5}$
 ${\displaystyle {\begin{bmatrix}1&2&3&\vdots &12\\2&3&1&\vdots &17\\3&1&2&\vdots &19\end{bmatrix}}}$ ${\displaystyle r_{2}-2r_{1}\Rightarrow r_{1}}$ ${\displaystyle {\begin{bmatrix}1&2&3&\vdots &12\\0&-1&-5&\vdots &-7\\3&1&2&\vdots &19\end{bmatrix}}}$ ${\displaystyle r_{3}-3r_{1}\Rightarrow r_{3}}$ ${\displaystyle {\begin{bmatrix}1&2&3&\vdots &12\\0&-1&-5&\vdots &-7\\0&-5&-7&\vdots &-17\end{bmatrix}}}$ ${\displaystyle r_{3}-5r_{2}\Rightarrow r_{3}}$ ${\displaystyle {\begin{bmatrix}1&2&3&\vdots &12\\0&-1&-5&\vdots &-7\\0&0&18&\vdots &18\end{bmatrix}}}$ ${\displaystyle tobecontinued}$

# Matrices

## Matrices

A matrix is a rectangular array of numbers enclosed in brackets. In a notational sense, what differentiates a list of numbers from a matrix is its format. The numbers are listed so that each number has a certain, specific position between the brackets. Each number, or value, in a matrix is called an entry.

One of the main benefits of matrices is the properties which allow them to be manipulated and used for many different, but useful purposes.

Matrices can vary in size. This variation in size is called dimensions. Just like the dimensions of a room (width x length) matrices have dimensions (number of rows x number of columns). Thus, a 2 x 3 (read 2 by 3) matrix will have 2 rows and 3 columns.

Example of a 2 x 3 matrix:

${\displaystyle M={\begin{bmatrix}572&98&302\\1&732&22\\\end{bmatrix}}}$

Another term associated with matrices is address. Like your home address, an address describes where each value, or entry, of a matrix lives. The address is composed of the lowercase letter of the matrix with the row and column number (in that order) as subscripts.

Using the 2 x 3 matrix M as an example, the positions of the values are as follows:

${\displaystyle M={\begin{bmatrix}m_{11}&m_{12}&m_{13}\\m_{21}&m_{22}&m_{23}\end{bmatrix}}={\begin{bmatrix}572&98&302&\\1&732&22\end{bmatrix}}}$
 ${\displaystyle m_{11}=572\quad m_{12}=98\quad m_{13}=302\quad m_{21}=1\quad m_{22}=732\quad m_{23}=22}$

A square matrix is any matrix that has the same number of rows as it does columns.

Example: 2 x 2 or 3 x 3 matrices are both square matrices.

${\displaystyle 2x2={\begin{bmatrix}{\color {red}1}&12\\21&{\color {red}2}\end{bmatrix}}\qquad 3x3={\begin{bmatrix}{\color {red}7}&9&30\\1&{\color {red}2}&2\\5&43&{\color {red}6}\end{bmatrix}}}$

Take note of the numbers in red above in the 2 x 2 and 3 x 3 square matrices. These numbers are in the addresses of the main diagonal. The main diagonal of a square matrix is the diagonal from the upper left corner entry to the bottom right corner entry. Notice that only square matrices can have a main diagonal.

## Adding and Subtracting Matrices

To add or subtract matrices, the sum or difference is found when addition or subtraction is applied to corresponding entries.

For example,

${\displaystyle {\begin{bmatrix}{\color {red}7}&{\color {blue}4}\end{bmatrix}}+{\begin{bmatrix}{\color {red}3}&{\color {blue}9}\end{bmatrix}}={\begin{bmatrix}{\color {red}10}&{\color {blue}13}\end{bmatrix}}}$
${\displaystyle {\begin{bmatrix}{\color {red}a_{11}}&{\color {blue}a_{12}}\end{bmatrix}}+{\begin{bmatrix}{\color {red}b_{11}}&{\color {blue}b_{12}}\end{bmatrix}}={\begin{bmatrix}{\color {red}a_{11}+b_{11}}&{\color {blue}a_{12}+b_{12}}\end{bmatrix}}}$
${\displaystyle {\begin{bmatrix}{\color {red}7}&{\color {blue}4}\end{bmatrix}}-{\begin{bmatrix}{\color {red}3}&{\color {blue}9}\end{bmatrix}}={\begin{bmatrix}{\color {red}4}&{\color {blue}-5}\end{bmatrix}}}$
${\displaystyle {\begin{bmatrix}{\color {red}a_{11}}&{\color {blue}a_{12}}\end{bmatrix}}-{\begin{bmatrix}{\color {red}b_{11}}&{\color {blue}b_{12}}\end{bmatrix}}={\begin{bmatrix}{\color {red}a_{11}-b_{11}}&{\color {blue}a_{12}-b_{12}}\end{bmatrix}}}$

Since addition or subtraction takes place using corresponding entries, matrices must have the same dimensions in order to complete either operation.

Consider this operation

${\displaystyle {\begin{bmatrix}52\\36\\\end{bmatrix}}+{\begin{bmatrix}12&16&5\\\end{bmatrix}}====>{\begin{bmatrix}a_{11}\\a_{21}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&b_{13}\\\end{bmatrix}}\quad {\color {red}CAN'T\quad BE\quad DONE}}$
${\displaystyle {\begin{bmatrix}a_{11}+b_{11}&?+b_{12}&?+b_{13}\\a_{21}+?&---&---\\\end{bmatrix}}}$

Now consider these matrices

${\displaystyle {\begin{bmatrix}52&4\\36&18\\\end{bmatrix}}+{\begin{bmatrix}12&16\\34&2\\\end{bmatrix}}====>{\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}}={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}\\a_{21}+b_{21}&a_{22}+b_{22}\\\end{bmatrix}}={\begin{bmatrix}64&20\\70&20\\\end{bmatrix}}}$

## Properties of Equality for Matrices

### Commutative Property

Matrix addition is commutative ${\displaystyle A+B=B+A}$

${\displaystyle {\begin{bmatrix}5&9&0\\1&3&2\\\end{bmatrix}}+{\begin{bmatrix}7&8&3\\4&7&6\\\end{bmatrix}}={\begin{bmatrix}7&8&3\\4&7&6\\\end{bmatrix}}+{\begin{bmatrix}5&9&0\\1&3&2\\\end{bmatrix}}}$

>______________ ${\displaystyle {\begin{bmatrix}12&17&3\\5&10&8\\\end{bmatrix}}={\begin{bmatrix}12&17&3\\5&10&8\\\end{bmatrix}}}$

### Associative Property

Matrix addition is associative. ${\displaystyle A+B+C=(A+B)+C=A+(B+C)}$

${\displaystyle {\begin{bmatrix}4&7\\1&8\\\end{bmatrix}}+{\begin{bmatrix}3&18\\24&7\\\end{bmatrix}}+{\begin{bmatrix}1&8\\14&6\\\end{bmatrix}}={\Bigg (}{\begin{bmatrix}4&7\\1&8\\\end{bmatrix}}+{\begin{bmatrix}3&18\\24&7\\\end{bmatrix}}{\Bigg )}+{\begin{bmatrix}1&8\\14&6\\\end{bmatrix}}={\begin{bmatrix}4&7\\1&8\\\end{bmatrix}}+{\Bigg (}{\begin{bmatrix}3&18\\24&7\\\end{bmatrix}}+{\begin{bmatrix}1&8\\14&6\\\end{bmatrix}}{\Bigg )}}$

${\displaystyle {\begin{bmatrix}4&7\\1&8\\\end{bmatrix}}+{\begin{bmatrix}3&18\\24&7\\\end{bmatrix}}+{\begin{bmatrix}1&8\\14&6\\\end{bmatrix}}={\Bigg (}{\begin{bmatrix}7&25\\25&15\\\end{bmatrix}}{\Bigg )}+{\begin{bmatrix}1&8\\14&6\\\end{bmatrix}}\qquad \quad ={\begin{bmatrix}4&7\\1&8\\\end{bmatrix}}+{\Bigg (}{\begin{bmatrix}4&26\\38&13\\\end{bmatrix}}{\Bigg )}}$

>_________${\displaystyle {\begin{bmatrix}8&33\\39&21\\\end{bmatrix}}\qquad \quad \quad =\qquad \qquad {\begin{bmatrix}8&33\\39&21\\\end{bmatrix}}\qquad \qquad \quad ={\begin{bmatrix}8&33\\39&21\\\end{bmatrix}}}$

The zero matrix is the additive identity matrix 'O'.

A Zero matrix is a matrix in which all of the entries are zero. ${\displaystyle O={\begin{bmatrix}0&0\\0&0\\\end{bmatrix}}}$

Any matrix added to the matrix 'O' will retain its' same values.

Example of additive identity ${\displaystyle A+O=A}$

${\displaystyle {\begin{bmatrix}7&4\\6&9\\\end{bmatrix}}+{\begin{bmatrix}0&0\\0&0\\\end{bmatrix}}={\begin{bmatrix}7&4\\6&9\\\end{bmatrix}}}$

It takes two matrices to form a pair of inverses. Two matrices are additive inverses if their sum is the zero matrix. This occurs when the additive inverse of a matrix contains the values opposite of each entry. ${\displaystyle A+(-A)=O}$

${\displaystyle {\begin{bmatrix}{\color {red}8}&{\color {red}2}&{\color {red}-1}\\{\color {red}-6}&{\color {red}9}&{\color {red}-3}\\\end{bmatrix}}+{\begin{bmatrix}{\color {blue}-8}&{\color {blue}-2}&{\color {blue}1}\\{\color {blue}6}&{\color {blue}-9}&{\color {blue}3}\\\end{bmatrix}}={\begin{bmatrix}{\color {red}8}+{\color {blue}-8}&{\color {red}2}+{\color {blue}-2}&{\color {red}-1}+{\color {blue}1}\\{\color {red}-6}+{\color {blue}6}&{\color {red}9}+{\color {blue}-9}&{\color {red}-3}+{\color {blue}3}\\\end{bmatrix}}={\begin{bmatrix}0&0&0\\0&0&0\\\end{bmatrix}}}$

### Multiplicative Identity Matrix

A multiplicative identity matrix, usually denoted by the letter I, is any square matrix that has a value of 1 in all the entries along the main diagonal and 0 in the remaining entries.

${\displaystyle I_{2x2}={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad I_{3x3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}}$

Any matrix multiplied by an identity matrix will retain it's original entries. ${\displaystyle AxI=A}$

${\displaystyle A={\begin{bmatrix}6&15&9\\23&5&43\\\end{bmatrix}}\quad \quad I={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}}$ ${\displaystyle }$

${\displaystyle AI={\begin{bmatrix}(6)(1)+(15)(0)+(9)(0)&(6)(0)+(15)(1)+(9)(0)&(6)(0)+(15)(0)+(9)(1)\\(23)(1)+(5)(0)+(43)(0)&(23)(0)+(5)(1)+(43)(0)&(23)(0)+(5)(0)+(43)(1)\\\end{bmatrix}}}$

${\displaystyle AI={\begin{bmatrix}6&15&9\\23&5&43\\\end{bmatrix}}}$

### Multiplicative Inverse Matrix

If matrices ${\displaystyle AxB=I}$ where I is the identity matrix, then A and B are multiplicative inverses of one another.

${\displaystyle A={\begin{bmatrix}-1&0&2\\4&1&-1\\2&0&1\\\end{bmatrix}}\quad \quad B={\begin{bmatrix}-0.2&0&0.4\\1.2&1&-1.4\\0.4&0&0.2\\\end{bmatrix}}}$

${\displaystyle AB={\begin{bmatrix}(-1)(-0.2)+(0)(1.2)+(2)(0.4)&(-1)(0)+(0)(1)+(2)(0)&(-1)(0.4)+(0)(-1.4)+(2)(0.2)\\(4)(-0.2)+(1)(1.2)+(-1)(0.4)&(4)(0)+(1)(1)+(-1)(0)&(4)(0.4)+(1)(-1.4)+(-1)(0.2)\\(2)(-0.2)+(0)(1.2)+(1)(0.4)&(2)(0)+(0)(1)+(1)(0)&(2)(0.4)+(0)(-1.4)+(1)(0.2)\\\end{bmatrix}}}$ ${\displaystyle AB=I={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}}$

Thus, matrices A and B are multiplicative inverses of each other.

## Multiplying Matrices

Another useful property of matrices is called a scalar. A scalar is a number located outside of a single matrix. To apply the scalar to the matrix, simply multiply each entry of the matrix by the scalar.

For example,

${\displaystyle {\color {red}3}{\begin{bmatrix}3&9&15\\1&7.5&2\\\end{bmatrix}}={\begin{bmatrix}({\color {red}3})3&({\color {red}3})9&({\color {red}3})15\\({\color {red}3})1&({\color {red}3})7.5&({\color {red}3})2\\\end{bmatrix}}={\begin{bmatrix}9&27&45\\3&22.5&6\\\end{bmatrix}}}$

In order to multiply two matrices together it is necessary to pay attention to their dimensions. Matrices A and J can be multiplied only if the number of columns in A equals the number of rows in J. Also, another hint is that the product of an a x j and a j x b matrix is an a x b matrix. Notice that the number of columns from the first matrix must equal the number of rows from the second matrix (j = j).

First, we'll look at how to test to see if we can multiply matrices.

Consider matrices Q and R.

${\displaystyle Q=2\quad x\quad {\color {red}3}\qquad \quad \qquad R={\color {red}3}\quad x\quad 2\qquad {\color {red}3\ columns}={\color {red}3\ rows}}$

Since the number of columns of matrix Q equals the rows of matrix R, these matrices can be multiplied. This will produce a 2 x 2 matrix.

${\displaystyle Q={\begin{bmatrix}{\color {red}1}&{\color {red}7}&{\color {red}9}\\3&5&1\\\end{bmatrix}}\qquad x\qquad R={\begin{bmatrix}{\color {red}8}&9\\{\color {red}3}&5\\{\color {red}4}&6\\\end{bmatrix}}}$

Multiplying Q and R.

During this process, you may find your fingers and mental addition extremely helpful. You can use your left pointer finger to follow the entries in the rows of matrix Q and your right pointer finger to follow the columns of matrix R. To multiply matrices, add the products of consecutive entries in corresponding rows of matrix Q and columns of matrix R.

${\displaystyle QR={\begin{bmatrix}{\color {red}q_{11}}&{\color {red}q_{12}}&{\color {red}q_{13}}\\{\color {red}q_{21}}&{\color {red}q_{22}}&{\color {red}q_{23}}\\\end{bmatrix}}{\begin{bmatrix}{\color {blue}r_{11}}&{\color {blue}r_{12}}&\\{\color {blue}r_{21}}&{\color {blue}r_{22}}\\{\color {blue}r_{31}}&{\color {blue}r_{32}}\\\end{bmatrix}}=}$ ${\displaystyle {\begin{bmatrix}QR_{11}={\color {red}q_{11}}\bullet {\color {blue}r_{11}}\quad +&{\color {red}q_{12}}\bullet {\color {blue}r_{21}}\quad +&{\color {red}q_{13}}\bullet {\color {blue}r_{31}}\quad QR_{12}={\color {red}q_{11}}\bullet {\color {blue}r_{12}}\quad +&{\color {red}q_{12}}\bullet {\color {blue}r_{22}}\quad +&{\color {red}q_{13}}\bullet {\color {blue}r_{32}}\\QR_{21}={\color {red}q_{21}}\bullet {\color {blue}r_{11}}\quad +&{\color {red}q_{22}}\bullet {\color {blue}r_{21}}\quad +&{\color {red}q_{23}}\bullet {\color {blue}r_{31}}\quad QR_{22}={\color {red}q_{21}}\bullet {\color {blue}r_{12}}\quad +&{\color {red}q_{22}}\bullet {\color {blue}r_{22}}\quad +&{\color {red}q_{23}}\bullet {\color {blue}r_{32}}\\\end{bmatrix}}}$

${\displaystyle QR={\begin{bmatrix}1&7&9\\3&5&1\\\end{bmatrix}}{\begin{bmatrix}8&9\\3&5\\4&6\\\end{bmatrix}}={\begin{bmatrix}(1)(8)+(7)(3)+(9)(4)&(1)(9)+(7)(5)+(9)(6)\\(3)(8)+(5)(3)+(1)(4)&(3)(9)+(5)(5)+(1)(6)\\\end{bmatrix}}={\begin{bmatrix}65&98\\43&58\\\end{bmatrix}}}$

## Determinants

Every square matrix has a value called a determinant, and only square matrices have defined determinants. The determinant of a 2x2 square matrix is the difference of the products of the diagonals.

The determinant of a 2 x 2 matrix can be found as follows:

${\displaystyle det{\begin{bmatrix}{\color {red}9}&{\color {blue}2}\\{\color {blue}7}&{\color {red}8}\\\end{bmatrix}}=({\color {red}9})({\color {red}8})-({\color {blue}7})({\color {blue}2})=58}$

The "down" diagonal is in red and the "up" diagonal is in blue. The up diagonals are always subtracted from the down diagonals. Matrices that are larger than a 2 x 2 matrix become a little more complicated when finding the determinant but the same rules apply.

• The "down" diagonal is not necessarily the same as the main diagonal mentioned earlier. The down diagonal happens to be the main diagonal for a 2 x 2 matrix but larger matrices will have multiple down diagonals and only one main diagonal.

Let's find ${\displaystyle det{\begin{bmatrix}1&6&19\\8&17&5\\14&9&3\\\end{bmatrix}}}$

When finding the determinant of a 3 x 3 matrix it is helpful to write the first two columns to the right side of the matrix like so,

${\displaystyle det{\begin{bmatrix}1&6&19\\8&17&5\\14&9&3\\\end{bmatrix}}{\begin{matrix}1&6\\8&17\\14&9\end{matrix}}}$

As shown above in the 2x2 matrix, the numbers are color coded. The blue numbers, once again, indicate they are used in the up diagonals, the red are used in the down diagonals, and those in magenta are used in both.

${\displaystyle det{\begin{bmatrix}{\color {red}1}&{\color {red}6}&{\color {magenta}19}\\8&{\color {magenta}17}&{\color {magenta}5}\\{\color {blue}14}&{\color {blue}9}&{\color {magenta}3}\\\end{bmatrix}}{\begin{matrix}{\color {blue}1}&{\color {blue}6}\\{\color {magenta}8}&17\\{\color {red}14}&{\color {red}9}\end{matrix}}=({\color {red}1}*{\color {red}17}*{\color {red}3})+({\color {red}6}*{\color {red}5}*{\color {red}14})+({\color {red}19}*{\color {red}8}*{\color {red}9})-{\Big [}({\color {blue}14}*{\color {blue}17}*{\color {blue}19})+({\color {blue}9}*{\color {blue}5}*{\color {blue}1})+({\color {blue}3}*{\color {blue}8}*{\color {blue}6}){\Big ]}}$

${\displaystyle =({\color {red}51}+{\color {red}420}+{\color {red}1,368})-({\color {blue}4,522}+{\color {blue}45}+{\color {blue}144})=-2872}$

Thus, the determinant of the above 3x3 matrix is -2872.

While square matrices of any size have a determinant, there is no way to extend this diagonal method of computing that determinant for a square matrix of size 4x4 or larger.

## Practice Problems

### Questions

Match the following terms with their definitions.

1. What is a matrix? A mold in which something, such as printing type or a phonograph record, is cast or shaped.

2. How can you change one format of an equation into another? If both funds are in the same fund" family" you can do an "Exchange"

3.How do you perform scalar multiplication? You just take a regular number called a "scalar" and multiply it on every entry in the matrix.

### Properties

Match the names of the properties with their equation equivalent.

____Additive Identity>______________1. ${\displaystyle A+B+C=(A+B)+C=A+(B+C)}$

____ Additive Inverse>______________2. ${\displaystyle AxI=A}$

____ Associative Property>__________3. ${\displaystyle A+B=B+A}$

____ Commutative Property>_________4. ${\displaystyle A+O=A}$

____ Multiplicative Identity>__________5. ${\displaystyle A+(-A)=O}$

____ Multiplicative Inverse>__________6. ${\displaystyle AxB=I}$

1) ${\displaystyle {\begin{bmatrix}52\\36\\\end{bmatrix}}+{\begin{bmatrix}12\\5\\\end{bmatrix}}}$

2) ${\displaystyle {\begin{bmatrix}12&7&17\\6&91&21\\\end{bmatrix}}+{\begin{bmatrix}0&14&32\\5&28&1\\\end{bmatrix}}}$

3) ${\displaystyle {\begin{bmatrix}3&65\\8&32\\\end{bmatrix}}+{\begin{bmatrix}9&71&16\\11&4&10\\\end{bmatrix}}}$

4) ${\displaystyle {\begin{bmatrix}4&56&14\\15&21&35\\\end{bmatrix}}+{\begin{bmatrix}87&13\\5&12\\\end{bmatrix}}}$

5) ${\displaystyle {\begin{bmatrix}2&34&41&23&14\\5&69&52&6&61\\\end{bmatrix}}+{\begin{bmatrix}1&4&5&17&53\\7&8&99&31&93\\\end{bmatrix}}}$

6) ${\displaystyle {\begin{bmatrix}19\\32\\44\\\end{bmatrix}}+{\begin{bmatrix}12\\89\\42\\\end{bmatrix}}}$

### Subtracting Matrices

Subtract the matrices.

1) ${\displaystyle {\begin{bmatrix}52\\36\\\end{bmatrix}}-{\begin{bmatrix}12\\5\\\end{bmatrix}}}$

2) ${\displaystyle {\begin{bmatrix}1&16&43\\6&21&89\\\end{bmatrix}}-{\begin{bmatrix}8&2&61\\9&17&26\\\end{bmatrix}}}$

3) ${\displaystyle {\begin{bmatrix}65&14\\4.5&43\\\end{bmatrix}}-{\begin{bmatrix}31&13\\0&24\\\end{bmatrix}}}$

4) ${\displaystyle {\begin{bmatrix}52&17&88\\2&63&16\\\end{bmatrix}}-{\begin{bmatrix}13&18\\4&11\\\end{bmatrix}}}$

5) ${\displaystyle {\begin{bmatrix}3.4&7.6\\5.2&9.6\\1.2&8.8\end{bmatrix}}-{\begin{bmatrix}9.8&7.6\\5.1&2.3\\6.9&4.5\\\end{bmatrix}}}$

### Multiplying Matrices

Multiply the matrices or by the scalar to find the product.

1) ${\displaystyle {\begin{bmatrix}4&7\\2&8\\3&9\\\end{bmatrix}}{\begin{bmatrix}0&6&9\\7&2&3\\\end{bmatrix}}}$

2) ${\displaystyle {\begin{bmatrix}4&7\\2&8\\3&9\\1&6\\\end{bmatrix}}{\begin{bmatrix}0&6&17&9\\7&2&36&12\\\end{bmatrix}}}$

3) ${\displaystyle {\begin{bmatrix}14&0&1\\4&2&9\\1&6&8\\\end{bmatrix}}{\begin{bmatrix}1&4&5\\1&2&7\\12&6&8\\\end{bmatrix}}}$

4) ${\displaystyle 8{\begin{bmatrix}41&17&54\\2&8&12\\4&5&3\\\end{bmatrix}}}$

5) ${\displaystyle {\begin{bmatrix}4&7\\2&8\\3&9\\\end{bmatrix}}{\begin{bmatrix}0&6\\7&2\\9&3\\\end{bmatrix}}}$

### Determinants

Find the determinant of the following matrices.

1) ${\displaystyle {\begin{bmatrix}5&1\\0&6\\\end{bmatrix}}}$

2) ${\displaystyle {\begin{bmatrix}1&12&4\\9&6&3\\2&2&0\\\end{bmatrix}}}$

3) ${\displaystyle {\begin{bmatrix}14&4&1\\1&2&7\\12&6&3\\\end{bmatrix}}}$

4) ${\displaystyle {\begin{bmatrix}1&4\\9&3\\\end{bmatrix}}}$

5) ${\displaystyle {\begin{bmatrix}7&0\\2&6\\\end{bmatrix}}}$

## Definitions Solutions

Match the following terms with their definitions.

8 address>_______________1. Diagonal from the upper left corner entry to the bottom right corner entry

6 determinant>____________2. A rectangular array of numbers enclosed in brackets

3 dimensions>____________3. Variation in size of a matrix

1 main diagonal>__________4. Any matrix that has the same number of rows as it does columns

2 matrix>________________5. Matrix in which all of the entries are zero

7 scalar>________________6. The difference of the products of the diagonals

4 square matrix>__________7. Number located outside of a single matrix which is multiplied by each entry of the matrix

5 zero matrix>____________8. Describes where each value, or entry, of a matrix lives

## Properties Solutions

Match the names of the properties with their equation equivalent.

4 Additive Identity>______________1. ${\displaystyle A+B+C=(A+B)+C=A+(B+C)}$

5 Additive Inverse>______________2. ${\displaystyle AxI=A}$

1 Associative Property>__________3. ${\displaystyle A+B=B+A}$

3 Commutative Property>_________4. ${\displaystyle A+O=A}$

2 Multiplicative Identity>__________5. ${\displaystyle A+(-A)=O}$

6 Multiplicative Inverse>__________6. ${\displaystyle AxB=I}$

## Adding Matrices Solutions

1) ${\displaystyle {\begin{bmatrix}64\\41\\\end{bmatrix}}}$

2) ${\displaystyle {\begin{bmatrix}12&21&49\\11&119&22\\\end{bmatrix}}}$

3)${\displaystyle \quad }$ Cannot be done because the matrices do not have the same dimensions.

4)${\displaystyle \quad }$ Cannot be done because the matrices do not have the same dimensions.

5) ${\displaystyle {\begin{bmatrix}3&38&46&40&67\\12&77&151&37&154\\\end{bmatrix}}}$

6) ${\displaystyle {\begin{bmatrix}31\\121\\86\\\end{bmatrix}}}$

## Subtracting Matrices Solutions

Subtract the matrices.

1) ${\displaystyle {\begin{bmatrix}40\\31\\\end{bmatrix}}}$

2) ${\displaystyle {\begin{bmatrix}-7&14&-18\\-3&4&63\\\end{bmatrix}}}$

3) ${\displaystyle {\begin{bmatrix}34&1\\4.5&19\\\end{bmatrix}}}$

4) Cannot be done because the matrices do not have the same dimensions.

5) ${\displaystyle {\begin{bmatrix}-6.4&0\\0.1&7.3\\-5.7&4.3\\\end{bmatrix}}}$

## Multiplying Matrices Solutions

Multiply the matrices or by the scalar to find the product.

1) ${\displaystyle {\begin{bmatrix}49&38&57\\56&28&42\\63&36&54\\\end{bmatrix}}}$

2) ${\displaystyle {\begin{bmatrix}49&38&320&120\\56&28&322&114\\63&36&375&135\\42&18&233&81\\\end{bmatrix}}}$

3) ${\displaystyle {\begin{bmatrix}26&62&78\\114&74&106\\103&64&111\\\end{bmatrix}}}$

4) ${\displaystyle {\begin{bmatrix}328&136&432\\16&64&96\\32&40&24\\\end{bmatrix}}}$

5) Cannot be done because the number of columns from the first matrix does not equal the number of rows in the second matrix.

## Determinants Solutions

Find the determinant of the following matrices.

1) 30

2) 90

3) -198

4) -33

5) 42

# Univariate Data

## univarite statistics

a basic introduction (not complete)

${\displaystyle \Sigma }$: "sigma" which means "sum of"

example:

${\displaystyle \sum _{i=2}^{4}1\times i}$

(considering numbers 1 to 10)

sum of numbers from 2 to 4 by factor of 1 (${\displaystyle 1\times i}$)

4

Σ 1xi = 2 + 3 + 4 = 9

i=2

the mean (${\displaystyle X}$) = sum of all scores / number of scores = Σx / n

also:

standard deviation (§)

variance (v)(§2)

box plots

-quartiles

stem plots

class limits, boundaries, classmarks

transformations on data sets

Standard scores (z)

# Sets

A set is a collection of items which have something in common. They are usually designated by upper case letters. The members of the set are called elements.

The number of elements in set A = n(A) or |A|

${\displaystyle 2\in A}$; 2 is an element of set A.

${\displaystyle 2\not \in C}$; 2 is not an element of set C.

${\displaystyle B\subset A}$; B is a subset of A which means every element of B is an element of A. ${\displaystyle B\not \subset A}$; B is not a subset of A.

U; universal set and varies for each collection under construction.

${\displaystyle \emptyset }$ is the empty set and has no elements in it. ${\displaystyle n(\emptyset )=0.}$

Operations on Sets:

${\displaystyle A\cap B}$; Intersection between sets: elements common to both sets.

${\displaystyle A\cap B=\emptyset }$; A and B are said to be disjoint if they have no common elements.

${\displaystyle A\cup B}$; The union between the sets A and B is the set of all the elements found in either A or B.

The complement of set A is the set of all elements in the universal set that are not found in A. A’ or . *Note: the complementation must be done in the context of the universal set.

For sets A and B; ${\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)}$

Venn Diagrams:

The relationship between sets and the operations performed on them can be represented graphically by using Venn Diagrams.

In Venn Diagrams, rectangles are usually used to represent universal sets and sets are usually defined on them represented by shaded circles or ovals.

Complement of a set:

Complement of set A is the set of all elements in the universal set NOT set A; A’ or .

Subsets:

Every element of a subset is also in the set. The set itself is also considered a subset. The empty set is also considered a subset.

Rules:

n(AB) = n(A) + n(B) – n(A  B)

n(ABC) = n(A) + n(B) + n(C) – n(A  B) – n(A  C) – n(B  C) + n(A  B  C)

# Transformations

A transformation of a given figure changes its size, position, or shape. Lines of reflection, angle measurements, and distances are used to describe transformations according to the type of transformation that occurred. Not only can you transform shapes and figures, but you can also transform functions. Piecewise functions can be transformed through the application of transformations to each piece individually. Different ways to transform functions are: horizontal and vertical translation, reflection across the x- and y-axis, and horizontal and vertical stretch/compression.

# Probability

Home - 1 - 2 - 3 - 4 - 5 - 6- 7 - 8 - 9 - 10

Probability is a way of expressing an expectation about the likelihood of an "event" occurring in an "experiment", based on whatever information is available either about the mechanism which lies behind the experiment (theoretical probability) or knowledge of previous events (experimental probability).

An example of a random experiment and often used in Maths puzzles is the birth of a child, in which the gender of the child cannot be known beforehand and is usually deemed to be 50/50 boy versus girl. In this case, the experiment is the birth, the possible "outcomes" are "girl" and "boy". The word "event" usually refers to the particular outcome we consider a "success" (which simply means the type of outcome whose probability we wish to calculate). Often, the "outcomes" are expressed as the result of a SERIES of experiments ("What are the chances of throwing heads twice in two tosses of the same coin").

## The Measure of Probability

The normal measure of probability is a number between 0 and 1, where 0 means "impossible" (not quite true, see below), .5 means "as likely as not" and 1 means "certain to happen". For example, the probability of tossing a head" is usually taken to be 0.5. In everyday language, the number is more likely to be expressed as a fraction (1/2 or "one in two") or a percentage ("50% chance"). When throwing a die, the probability of getting a 1 is 1/6 (i.e. we expect that in a large number of throws, the frequency of "1s" will be close to 1/6 of the number of throws).

## Two kinds of Probability

• Theoretical Probability - Based on a knowledge of the mechanism behind the event. It is simplest to calculate when there are a finite number of possible outcomes, all believed to be equally likely. The probability of an event is simply the number of outcomes which would match our "event", divided by the total number of possible outcomes. On a die, there are 6 possible (and equally-likely) outcomes for each roll, but only one of these outcomes results in a 1. Therefore we calculate the probability of rolling a 1 as 1/6. This is the most common type of probability used in math classes. The usual mistake made in calculating probability this way is not checking carefully whether you are counting "equally likely" outcomes. (To be silly, one might say "there are two possible outcomes ; 1 or not 1, therefore the chance of rolling a 1 is 1/2"). The more common reason for falsely believing asset of outcomes to be equally likely is either that the dice or coin is "loaded" or that you have missed a possible outcome, or confused two separate outcomes as a single one. For instance, if the coin is thick enough, the possibility that the coin can land stably on its edge should not be ignored. When tossing TWO coins, many people think there are three equally likely outcomes (two heads, two tails, one of each) and that therefore the chance of getting "one of each" is 1/3: but actually there are two different ways of getting a head and a tail, so the chance of getting "one of each" is actually 2/4 = 50%. If you are sure that you have calculated the probability correctly then you can ascribe a single number to the probability of the event, but the only way to be sure that you have not missed a possibility, or done the sums wrong, is to go ahead and try it lots of time and see what happens. But you are then into "Experimental Probability".
• Experimental Probability - Based on experience. This is less prone to mistakes (provided the experimental conditions are fair) and easy to calculate the result (number of successful trials divided by total number of trials). The larger the number of times that you perform the test, the closer you will come to the theoretical probability (provided that you calculated the theoretical probability correctly!), but the greater the chance that the result will not be EXACTLY the same.

In Maths classes, the first type of probability is more common, in science, the second.

## Meaning of Probability

### Repeated Events

Where an event can be repeated in the same way as often as desired, then not only is it clear what the calculated probability means ("I expect that if I toss this coin 500 times, I will see about 250 heads"), but also it will be possible to check whether the estimate was correct (toss the coin and see how many times you get a head).

## One-offs

Where a trial (or event) is a one-off (neither this trial, nor a very similar one can be done again) then not only must the calculation be a theoretical one, but also it is less clear what the calculated figure MEANS. "There is a 50% chance of the Earth being destroyed by collision with a comet during this year". Either the Earth will or will not be destroyed, neither outcome will show you whether your calculated probability was correct. Actually, you CAN run the trial as many times as you like (treat each year as one trial), but there will be a maximum of ONE "successful" outcome, so it is not easy to see what light 1 or 2 or 3 or 4 destruction-free years sheds on your calculation. If, however, you have the opportunity to gamble on many different one-off events, and experience gives you good reason to trust your probability estimates, then probability does have some meaning: you would be wise in future wagers to bet on events to which you ascribe high probability and better to avoid improbable ones (except at very favorable odds). This is actually similar to horse-racing - you are encouraged to believe you can calculate the odds of a horse winning, but most races are really one-offs run in very different circumstances to all the other races. If you only bet on one race in your life, then you are unlikely to back a real outsider at whatever odds - but if you spend a lot of time at racecourses, you may probably back quite a lot of outsiders, but only in the cases where you believe that the high odds against them are more generous then necessary - expecting (perhaps unwisely, since it is the bookies, not you, who set the odds) that the few that win will pay enough to outweigh your many small losses on the ones that did not.

## Outcomes and sample spaces

When doing experiments, the answer is not known prior to the actual experiment. However, we should know what we can get. The possible results of an experiment are called outcomes. A certain group of outcomes is called a sample space.

When you throw a die, you can get 1, 2, 3, 4, 5 or 6 eyes. These are the possible outcomes in a throw. the sample space is U = {1, 2, 3, 4, 5, 6}. When we're looking at whether a birth results in boy(B) or girl(G), the sample space is U = {B, G}

## Model of probability

A model of probability gives the probability of every outcome a number between 1 and 0. The sum of the probability of all possible outcomes is 1 (if you cannot make them add up to 1, then you have missed or double-counted some possibilities).

Let's for instance take a die-throw as an example. The model of probability for that would be:

P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

However, when we're looking at whether a birth results in a boy or a girl, the model of probability results in:

P(B) = 0.514 and P(G) = 0.486

In the first example, all the outcomes have the same probability - the model is uniform. In the second example, the outcomes have different probabilities - the model is not uniform.

# Counting Techniques

## Factorials

A factorial, symbolized by n! with n being a number between 1 and infinite, is a product of consecutive counting numbers starting at 1 and ending at n. The number 0 is an exception. 0! has a value of 1. Factorials are used in many ways, from their use in equations to solving word problems. Their most useful application is through the calculation of the number of outcomes of an event. For example, it can produce the possible outcomes of the race, and how many combinations of first, second, third, and so forth of the racers.

n Factorial

For any whole number n, it's factorial is the product of the natural numbers that are less than or equal to it. For example:

7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040

The algebra behind factorials looks like this:

n! = n x (n-1) x (n-2) x (n-3) x ... x 1

## Permutations

Permutations is all the possible outcomes of a given value. For example, if you were given three colors (red [r], green [g], and yellow [y]) and were asked to find possible arrangements of them, you would find 6 possible arrangements: rgy, ryg, gyr, gry, yrg, ygr.

In a permutation, the order of the objects in the group is important. Using the same example above, we are given three different colors (red [r], green [g], and yellow [y]). If we were just given red [r], then there would be only one way to arrange it: red. However, when green [g] is thrown in the mix, there are now two ways to arrange these two colors: rg or gr. And now, when a third color is thrown in (yellow [y]), then there are now six possible outcomes, as stated above. The amount of permutations of the 3 colors is equal to 3 x 2 x 1, which equals 6. This is also equal to the factorial of 3:

3! = 3 x 2 x 1 = 6

The general formula for a permutation of n items is:

n x (n - 1) x (n - 2) x (n - 3) x ... x 1

As mentioned above, this is also called the n factorial, and is written like this:

n!

But, what if you don't want to order the whole set of things? Let's say that you need to select an order of 3 things from a group of 8. The Fundamental Counting Principle is one way to do just that.

First Thing: 8 possibilities

Second Thing: 7 possibilities

Third Thing: 6 possibilities

The number of permutations in this example is 336. In other words, there are 8 things, and we are choosing 3 of them in order. The 336 comes from the 8 possibilities x 7 possibilities x 6 possibilities.

Factorials, as stated above, can also be used to find permutations. The number of arrangements can be divided by the number of arrangements not used. Using the example above, there are 8 things, 3 arrangements of which do not matter. For large numbers of objects, it is useful to use this generalized formula:

nPr = n! / (n-r)!

This algebra, stated in words, reads like this: the number of permutations of n items taken r at a time is ___ .

So, in the example used above, 8 things taken 3 at a time would give us:

8P3 = 8! / (8-3)! = 8! / 5! = 336

## Permutation Examples

1. How many ways could all the students in a high school vote for the homecoming king and queen from a group of 6 people?

In other words, this question is asking us to select and arrange 2 things from 6 things. So, first thing to do isd to substitute 6 in for n in our nPr formula and 2 in for r:

6P2 = 6! / (6-2)! = 6! / 4! = (6 x 5 x 4 x 3 x 2 x 1) / (4 x 3 x 2 x 1) = 6 x 5 = 30

So, there are 30 ways to vote for the homecoming king and queen.

2. You have 10 picture frames to hang on your bedroom wall; however, you only have enough space for 5 frames. How many ways can you hang any 5 picture frames on your wall?

10P5 = 10! / (10-5)! = 10! / 5! = (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (5 x 4 x 3 x 2 x 1) = 10 x 9 x 8 x 7 x 6 = 30,240

So, there are 30,240 ways that you can pick the pictures you want to hang on your wall.

If you notice, in the above two examples, the number of things selected is equal to the number of factors left after dividing the two factorials. For instance, in example 1, there were 2 people we were voting for and, thus, 2 factors that were left after dividing the factorials: 6 x 5.

## Combination

When items are grouped and the order in which they are grouped doesn't matter, this is known as a combination. Normally, when order doesn't matter, there are less ways to choose things. Let's say we have 4 different letters: A, B, C, and D. There are 24 different ways to put these 4 letters in order (permutation). However, they are all the same combination for letters.

24 permutations: ABCD, ACDB, ADBC, ACBD, ADCB, ABDC, BACD, BADC, BDCA, BDAC, BCAD, BCDA, CDAB, CDBA, CBAD, CBDA, CABD, CADB, DABC, DACB, DBCA, DBAC, DCAB, DCBA

1 combination: ABCD

The formula previously used for permutations can be changed and used to find the number of combinations:

(ways to arrange all of items) / (ways to arrange items not selected) = number of permutations

To change this formula to find the number of combinations, the number of permutations will be divided by the number of ways to arrange the items selected. This is due to the fact that in combinations, order does not matter.

(ways to arrange all of items) / (ways to arrange items selected) x (ways to arrange items not selected) = number of combinations

So, the algebraic formula for finding combinations of n items taken r at a time is:

nCr = n! / r! x (n - r)!

For example, the number of combinations of 8 items taken 4 at a time is:

8C4 = 8! / 4!(8-4)! = 70

So, how do you know whether or not to use a permutation versus a combination. The first thing you need to figure out is whether or not order matters. If it does, then permutations are needed to solve the problem. However, if order does not matter, then combination is the method to be used.

## Combination Examples

1. Lauren wants to buy 4 puppies for her farm. The puppy store has 12 puppies to choose from. How many ways can she choose 4 puppies? The first thing to do is decide whether to use the formula for permutation of combination to solve the problem. Since order is not an issue, the combination method is to be used.

The combination method looks like this:

12C4 = 12! / 4!(12-4)! = 12! / 4!(8!)

= (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (4 x 3 x 2 x 1)(8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)

= (12 x 11 x 10 x 9) / (4 x 3 x 2 x 1) = 495

So, there are 495 different ways to choose 4 puppies from a given 12 puppies.

# Bivariate Data

## Bivariate Data

Bivariate Data is data of two quantitative variables. This kind of data is analogous to what is known as univariate data, which is data of one quantitative variable (which could also have been deduced from their prefixes: "bi" means two and "uni" means one).

Bivariate data is used mainly in statistics. It is used in studies that include measures of central tendencies (i.e. mean, median, mode, midrange), variability, and spread. In each study, more than one variable is collected, like in medical studies. Height and weight of individuals might want to be obtained, not just height or weight. Bivariate data is the comparing of two quantitative variables of an individual, and can be shown graphically through histograms, scatterplots, or dotplots.

## Example

Based on the data given below, do women generally marry at a younger age than men do?

In this example, we are given two variables: gender and age. The data is given to us below:

10 men's and women's ages of when they were married

Men:

25, 26, 27, 29, 30, 31, 33, 36, 38, 40

Women:

19, 20, 21, 22, 23, 25, 26, 28, 29, 30

With this data, you can create a histogram to graphically see the results and how they relate to one another. Then, find the mean of each separate chart. The mean age of when men get married is:

25 + 26 + 27 + 29 + 30 + 31 + 33 + 36 + 38 + 40

10

= 31.5

The mean age of when women get married is:

19 + 20 + 21 + 22 + 23 + 25 + 26 + 28 + 29 + 30

10

= 24.3

Both distributions are slightly skewed to the right, meaning that more of the data values occur to the left of the mean than to the right.

So, based on our data, women do typically marry at a younger age than men do.

# Linear Programming and Graphical Solutions

## Linear Programming

Linear programming is a method that is used to find a minimum or maximum value for a function. That value is going to satisfy a known set of conditions constraints. Constraints are the inequalities in the linear programming problem. Their solution is graphed as a feasible region, which is a set of points. These points are where the graphs of the inequalities intersect. And, the region is said to be bounded when the graph of a system of constraints is a polygonal region.

There are 7 steps that are used when solving a problem using linear programming.

1. Define the variables.

2. Write a system of inequalities.

3. Graph the system of inequalities.

4. Find the coordinates of the vertices of the feasible region.

5. Write a function to be maximized or minimized.

6. Substitute the coordinates of the vertices into the function.

7. Select the greatest or least result. Answer the problem.

Many types of real-world problems can be solved using linear programming. These problems have restrictions placed on the variables. Some function of the variable must be maximized or minimized.

One Company sells two different products A and B, making a profit of Rs.40 and Rs.30 per unit, respectively. They are both produced with help of common production process and are sold in two different markets. The production process has a total capacity of 30000 man - hours. it takes 3 hours to produce an unit of A and one hour to produce an unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 8000 units and that of B is 12000 units. Subject to those limitations; products can be sold in any combination. Formulate this problems as an Linear Programming Model to maximize profit.

## Objective Function

Sometimes, in linear programming problems, identifying the feasible region is not the only thing necessary to be done. To maximize or minimize the function, the best combination of values must be found. This is what the objective function does. It can either have a maximum value, a minimum value, both, or neither. It all depends upon the feasible region.

There are two different general types of regions: bounded and unbounded regions. Bounded feasible regions have both a minimum and a maximum value. Unbounded feasible regions have either a minimum or maximum value, never both. The minimum or maximum value of such objective functions always occurs at the vertex of the feasible region. This mathematical idea, however, is a proof that is for more advanced mathematics.

## Bounded Region Example

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x,y) = 3x + y for this region. x > 1

y > 0 2x + y < 6

1. Find the vertices of the region. Graph the inequalities.

The polygon formed is a triangle with vertices at (1, 4), (3, 0), and (1, 0).

2. Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function. The maximum value is 9 at (3, 0). The minimum value is 3 at (1, 0).

## Unbounded Region Example

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y)= 5x + 4y for this region.

2x + y > 3

3y - x < 9

2x + y < 10

Graph the system of inequalities. There are only two points of intersection: (0, 3) and (3, 4).

The maximum is 31 at (3, 4).

Although f(0, 3) is 12, it is not the minimum value since there are other points in the solution that produce lesser values. For example, f(3, -2) = 7 and f(20, -35) = -40. It appears that way because the region is unbounded, f(x, y) has no minimum value.

## Maximum and Minimum Values

Sometimes, it is necessary to find the minimum or maximum value that a linear function has for the points in a feasible region. The minimum or maximum value of a related function always occurs at one of the vertices of the feasible region.

# Probability Distributions

## Probability Distribution

Some experiments have numerical outcomes, like rolling a die for instance. A variable whose value is the numerical outcome of a random event is called a random variable. Take rolling a die, for example. We can let the random variable D represent the number showing on the die when rolling the die. Then, D equals either 1, 2, 3, 4, 5, or 6.

A function that puts together a probability with its outcome in an experiment is known as a probability distribution. Or, another way of putting it is that it is a function that maps the sample space to the probabilities of the outcomes in the sample space for a particular random variable. The numbers below illustrate the probability distribution for rolling a die.

D = roll 1 2 3 4 5 6

Probability 1/6 1/6 1/6 1/6 1/6 1/6

P(D = 4) = 1/6

A uniform distribution is a distribution where all of the probabilities are the same. The probability distribution above has uniform distribution.

The use of a table of probabilities (or a graph) can help you visualize a probability distribution. Such graphs are known as relative- frequency histograms.

## Example of Probability Distribution

Suppose 2 dice are rolled. The table shows the distribution of the sum of the numbers rolled.

S = Sum 2 3 4 5 6 7 8 9 10 11 12 Probability 1/36 1/18 1/12 1/9 5/36 1/6 5/36 1/9 1/12 1/18 1/36

a. Use the table to find P(S = 10). What other sum has the same probability?

The probability of the sum of 9 (according to the table) is 1/12. The other sum of 4 has the same probability of 1/12.

b. What are the odds of rolling a sum of 8?

Step 1 Identify s and f.

             P(rolling an 8) = 5/36

= s/(s + f)     s = 5, 36－5=31， f = 31


Step 2 Find the odds.

              Odds = s:f

                   = 5:31


So, the odds of rolling a sum of 8 are 5:31.

# Odds

"Odds" is a way of expressing the likelihood of an event.

The more usual way of expressing the likelihood of an event is its "probability" (the percentage of future trials which are expected to produce the event: so in tossing a coin believed to be fair, we would assign a probability of 50% (or one half, or 0.5) to the event "heads").

The ODDS of an event, however, is the ratio of the probability of the event happening to the probability of the even not happening (i.e. the ODDS of a fair coin landing heads is 50%:50% = 1:1 = 1). It is the ODDS we are using when we use a phrase like "it is 50/50 whether I get the job" or "The chances of our team winning are 2 to 1".

Or, For example, when rolling a fair die, there is one chance that you will roll a 1 and five chances that you will not. The odds of rolling a 1 are 1:5, or 1 to 5. This can also be expressed as ${\displaystyle 1/5}$ or 0.2 or 20%, but these forms are likely to be misunderstood as normal probabilities rather than odds.

To convert odds to probability, you add the two parts, and this is the denominator of the fraction of your probability. The first part becomes the numerator. Thus, 1:5 becomes 1/(1+5) or 1/6.

To convert probability to odds, you use the numerator as the first number, then subtract the numerator from the denominator and use it as the second number. Thus, 1/6 becomes 1:(6-1) or 1:5.