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13.1 Solving linear equations

13.1 Solving linear equations

In earlier work on solving equations, you may have noticed there can be more than one way to solve
an equation. You can use any method you prefer, as long as it works. You should write out each step in
your solution neatly and check your answer at the end.

Worked example 13.1

Solve the equation 2(x − 5) = 2 + 8x.

First method
2(x − 5) = 2 + 8x
A 2x − 10 = 2 + 8x Multiply out the brackets.
A −10 = 2 + 6x Subtract 2x from each side.
A −12 = 6x Subtract 2 from each side.
A −2 = x Divide each side by 6.
Second method
2(x − 5) = 2 + 8x
x − 5 = 1 + 4x
A Divide each side by 2.
A x − 6 = 4x Subtract 1 from each side.
A −6 = 3x Subtract x from each side.
A −2 = x Divide each side by 3.
Check the answer:
x = −2 A 2(x − 5) = 2 × (−2 − 5) = 2 × −7 = −14 Both sides of the equation have the same value, −14.
and
x = −2 A 2 + 8x = 2 + (8 × −2) = 2 + −16 = −14 There are other ways you could solve this equation.
For example, in the first method you could subtract
8x instead of 2x and get −6x − 10 = 2. You should
get the same answer.

) Exercise 13.1

1 Solve these equations.
a 4x + 8 = 14 b 4x + 14 = 8 c 4x + 14 = −8 d −4x + 8 = 14

2 Solve these equations.
a a + 15 = 4 b a + 15 = 4a c a + 15 = 4a − 3 d a − 15 = 4a + 3
3 Solve these equations.
a 12 − y = 4 b 12 − y = −4 c 12 − 2y = 4 d 12 − 2y = −4

4 Solve these equations. Check each of your answers by substitution.
a 6 = 2d − 4 b 6 = 2(d − 4) c 6d = 2d − 4 d 6d = 2(d − 4)

5 Here is an equation.
2(x + 12) = 4x − 6
a Solve the equation by first multiplying out the brackets.
b Solve the equation by first dividing both sides by 2.

6 Solve these equations. Check your answers.
a 5 + 3x = 3 + 5x b 5 + 3x = 3 − 5x c 5 − 3x = 3 − 5x

13 Equations and inequalities 125