# Basic Algebra/Systems of Linear Equations/Solving Linear Systems by Substitution

## Lesson

To solve a system of linear equations without graphing, you can use the substitution method. This method works by solving one of the linear equations for one of the variables, then substituting this value for the same variable in the other linear equation and solving for the other variable. It does not matter which equation you choose first, or which variable you solve for first; the values for both variables will be the same.

## Example Problems

For example, given the system of linear equations:

${\displaystyle 2x-3y=-2}$

${\displaystyle 4x+y=24}$

The first step would be to choose one of the equations and solve it for either x or y. In the second equation y is not multiplied by a constant so it can be isolated in fewer steps.

${\displaystyle 4x+y=24}$

${\displaystyle y=24-4x}$

Now you have a value, ${\displaystyle 24-4x}$ , for y. Substitute this value of y into the first equation.

${\displaystyle 2x-3y=-2}$

${\displaystyle 2x-3(24-4x)=-2}$

Solve this equation for the variable x.

${\displaystyle 2x-3(24-4x)=-2}$

${\displaystyle 2x-72+12x=-2}$

${\displaystyle 14x-72=-2}$

${\displaystyle 14x=70}$

${\displaystyle x=5}$

We now have a value for x that can be substituted into either equation to solve for y.

${\displaystyle 4x+y=24}$

${\displaystyle 4(5)+y=24}$

${\displaystyle 20+y=24}$

${\displaystyle y=4}$

The solution to this system of linear equations is ${\displaystyle x=5}$ , ${\displaystyle y=4}$ . This can also be written as ${\displaystyle (x,y)=(5,4)}$ .

NOTE: If we substitute the value of x into the other equation, the value of y will remain the same.

${\displaystyle 2x-3y=-2}$

${\displaystyle 2(5)-3y=-2}$

${\displaystyle 10-3y=-2}$

${\displaystyle -3y=-12}$

${\displaystyle y=4}$

## Practice Games

[1] Practice Problems

[2] Math Drills

## Practice Problems

Note: Use `/` as the fraction line and put spaces between wholes and fractions!

Solve the system of linear equations.

1 ${\displaystyle 7x+2y=16}$

${\displaystyle -21x-6y=24}$

 Number of solutions:

2 ${\displaystyle 9x+4y=20}$

${\displaystyle 8x-10y=60}$

 x= , y=

3 ${\displaystyle t-2r=3}$

${\displaystyle 5r-4t=6}$

 r= , t=

4 ${\displaystyle 3v-w=7}$

${\displaystyle 2v+3w=1}$

 v= , w=

5 ${\displaystyle p-4q=-21}$

${\displaystyle 3p+q=2}$

 p= , q=