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

Manipulating Equations

Collecting Like Terms

Simplify the following:

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

Multiplication

Simplify the following:

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

Fractions

Simplify the following:

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

Solving Equations

Changing the Subject of an Equation

1. Solve for $x$:

$y=2x$

2. Solve for $z$:

$x=3z+8$

3. Solve for $y$:

$b={\sqrt {y}}$

4. Solve for $x$:

$y=x^{2}-9$

5. Solve for $b$:

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

Solving Quadratic Equations

Find the Roots of:

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

Simultaneous Equations

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 $a$ and a single as $s$, the two equations would be:

$2a+s=10$

$a+2s=8$

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 $c$ and a bottle of drink as $d$, the two equations would be:

$5c+8d=10$

$10c+6d=10$

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 $g$ and a gummi bear as $b$, the two equations would be:

$b+5=g$

$8b+9g=164$