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

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) $$8x^2 + 7xy + x^2 - 10x^2 + 4x^2y - 4xy^2$$

Multiplication
Simplify the following:


 * 1) $$2x \times 2x$$
 * 2) $$6xy \times 3xy$$
 * 3) $$6zb \times 3x \times 2ab$$
 * 4) $$3x^2 \times 4xy^2 \times 5x^2y^2z^2$$
 * 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) $$\frac{x}{y} + \frac{y}{x}$$

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:

$$b+5=g$$

$$8b+9g=164$$

Equations/Problems