A-level Mathematics/OCR/C1/Equations/Solutions

Manipulating Equations

Collecting Like Terms

1.

Here we only have $x$ terms, so we can add them together:

$x+x=2x$

2.

We have a similar situation to problem 1, but this time with $x^{2}$ terms. We can combine them by adding their coefficients:

$x^{2}+3x^{2}=(1+3)x^{2}=4x^{2}$

3.

Here we have three different terms in the mix ($x$, $x^{2}$ and $x^{3}$) so we need to be a little more careful. We can only combine coefficients if they represent the same term. Separate the expression into terms of $x$, $x^{2}$ and $x^{3}$ as follows:

$(3x+2x)+(2x^{2}-2x^{2})+3x^{3}$

Now combine the coefficients of the like terms:

$(3+2)x+(2-2)x^{2}+3x^{3}=5x+3x^{3}$

4.

This is another case with three distinct terms ($zy$, $z$ and $y$). Combine the coefficients of like terms:

${\begin{aligned}zy+2zy+2z+2y&=(1+2)zy+2z+2y\\&=3zy+2z+2y\end{aligned}}$

5. $-x^{2}+4x^{2}y-4xy^{2}+7xy$

Multiplication

1. $4x^{2}$
2. $18x^{2}y^{2}$
3. $36ab^{2}xz$
4. $60x^{5}y^{4}z^{2}$
5. ${\sqrt {x^{5}}}$

Fractions

1. $x$
2. ${\frac {7x}{12}}$
3. ${\frac {16xy}{15}}$
4. $2x-y+z$
5. ${\frac {x^{2}+y^{2}}{xy}}$

Solving Equations

Changing the Subject of an Equation

1. Solve for x.

$x={\frac {y}{2}}$

1. Solve for z.

$z={\frac {x-8}{3}}$

1. Solve for y.

$y=b^{2}$

1. Solve for x.

$x={\sqrt {y+9}}$

1. Solve for b.

$b={\frac {6y+7z}{6}}$

Solving Quadratic Equations

Find the Roots of:

1. $x=-2\ or\ 3$
2. $x=7\ or\ {\frac {3}{2}}$
3. $x=2\ or\ 3$
4. $x=-1\ or\ 0$
5. $x=-3\ or\ 4$

Simultaneous Equations

1. $a=4,s=2$
2. $c={\frac {2}{5}},d=1$
3. $b=7,g=12$