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

Simplify the following:

1. ${\displaystyle x+x}$
2. ${\displaystyle x^{2}+3x^{2}}$
3. ${\displaystyle 3x+2x^{2}+2x-2x^{2}+3x^{3}}$
4. ${\displaystyle zy+2zy+2z+2y}$
5. ${\displaystyle 8x^{2}+7xy+x^{2}-10x^{2}+4x^{2}y-4xy^{2}}$

Simplify the following:

1. ${\displaystyle 2x\times 2x}$
2. ${\displaystyle 6xy\times 3xy}$
3. ${\displaystyle 6zb\times 3x\times 2ab}$
4. ${\displaystyle 3x^{2}\times 4xy^{2}\times 5x^{2}y^{2}z^{2}}$
5. ${\displaystyle x^{2}{\sqrt {x}}}$

Simplify the following:

1. ${\displaystyle {\frac {x}{2}}+{\frac {x}{2}}}$
2. ${\displaystyle {\frac {x}{3}}+{\frac {x}{4}}}$
3. ${\displaystyle {\frac {3xy}{15}}-{\frac {xy}{3}}+{\frac {6xy}{5}}}$
4. ${\displaystyle {\frac {4x}{2}}-{\frac {4y}{4}}+{\frac {8z}{8}}}$
5. ${\displaystyle {\frac {x}{y}}+{\frac {y}{x}}}$

1. Solve for ${\displaystyle x}$:

${\displaystyle y=2x}$

2. Solve for ${\displaystyle z}$:

${\displaystyle x=3z+8}$

3. Solve for ${\displaystyle y}$:

${\displaystyle b={\sqrt {y}}}$

4. Solve for ${\displaystyle x}$:

${\displaystyle y=x^{2}-9}$

5. Solve for ${\displaystyle b}$:

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

Find the Roots of:

1. ${\displaystyle x^{2}-x-6=0}$
2. ${\displaystyle 2x^{2}-17x+21=0}$
3. ${\displaystyle x^{2}-5x+6=0}$
4. ${\displaystyle x^{2}+x=0}$
5. ${\displaystyle -x^{2}+x+12=0}$

Example 1

At a record store, 2 albums and 1 single costs £10. 1 album and 2 singles cost £8. Find the cost of an album and the cost of a single.

Taking an album as ${\displaystyle a}$ and a single as ${\displaystyle s}$, the two equations would be:

${\displaystyle 2a+s=10}$

${\displaystyle a+2s=8}$

You can now solve the equations and find the individual costs.

Example 2

Tom has a budget of £10 to spend on party food. He can buy 5 packets of crisps and 8 bottles of drink, or he can buy 10 packets of crisps and 6 bottles of drink.

Taking a packet of crisps as ${\displaystyle c}$ and a bottle of drink as ${\displaystyle d}$, the two equations would be:

${\displaystyle 5c+8d=10}$

${\displaystyle 10c+6d=10}$

Now you can solve the equations to find the cost of each item.

Example 3

At a sweetshop, a gobstopper costs 5p more than a gummi bear. 8 gummi bears and nine gobstoppers cost £1.64.

Taking a gobstopper as ${\displaystyle g}$ and a gummi bear as ${\displaystyle b}$, the two equations would be:

${\displaystyle b+5=g}$

${\displaystyle 8b+9g=164}$