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

< A-level Mathematics‎ | OCR‎ | C1‎ | Equations

## Manipulating Equations

### Collecting Like Terms

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

### Multiplication

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

### Fractions

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.

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

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:

$b+5=g$

$8b+9g=164$