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

Manipulating Equations

Collecting Like Terms

1. x + x
2. x² + 3x²
3. 3x + 2x² + 2x − 2x² + 3x³
4. zy + 2zy + 2z + 2y
5. 8x² + 7xy + x² − 10x² + 4x²y − 4xy²

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 = x² − 9

5. Solve for b.

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

Solving Quadratic Equations

Find the Roots of:

1. x² − x − 6 = 0
2. 2x² − 17x + 21 = 0
3. x² − 5x + 6 = 0
4. x² + x = 0
5. − x² + 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