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

1.

Here we only have ${\displaystyle x}$ terms, so we can add them together:

${\displaystyle x+x=2x}$

2.

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

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

3.

Here we have three different terms in the mix (${\displaystyle x}$, ${\displaystyle x^{2}}$ and ${\displaystyle x^{3}}$) 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 ${\displaystyle x}$, ${\displaystyle x^{2}}$ and ${\displaystyle x^{3}}$ as follows:

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

Now combine the coefficients of the like terms:

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

4.

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

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

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

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

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

1. Solve for x.

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

1. Solve for z.

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

1. Solve for y.

${\displaystyle y=b^{2}}$

1. Solve for x.

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

1. Solve for b.

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

Find the Roots of:

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

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