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Substitution

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How to substitute numbers into expressions and formulae, including negative numbers, squares and fraction bars, and work out the value in the right order.

What substitution means

Substitution means putting numbers in place of the letters in an expression or formula, then working out its value. For example, when x = 4, the expression x + 9 has the value 4 + 9 = 13

A number written next to a letter means multiply. When x = 7, 4x means 4 \times 7 = 28. Two letters written together multiply too: mn means m \times n

In the exam you are usually given a formula, such as W = 5m + 2n, and asked to “Work out the value of W” for given values of the letters. It is usually worth 2 marks: one for writing the numbers in place of the letters, and one for the answer. Harder questions use negative numbers, squares or a fraction.

Write a substitution line

Write the expression again with each letter replaced by its number. Put a \times sign wherever a number meets a letter, and put brackets round every negative number. When k = -2, 5k + 1 becomes 5 \times (-2) + 1

Then work it out in the usual order: brackets first, then powers, then multiply and divide, and add and subtract last. So 3 + 4 \times 2 = 3 + 8 = 11

Writing the substitution line earns a mark on its own, even if a later step goes wrong. On a calculator, type the brackets round a negative number as well.

Worked example 1: One letter, a positive number

Work out the value of 5x - 3 when x = 4

  1. Replace x with 4: 5 \times 4 - 3

  2. Your turn. Multiply first. What is 5 \times 4?

    5 \times 4 = 20

  3. Then subtract: 20 - 3 = 17
  4. Answer: 17

Multiplying by a negative number

When you substitute a negative number, you usually have to multiply by it. Keep the minus sign with the number by writing the number in brackets.

  • A positive number times a negative number is negative: 4 \times (-3) = -12
  • A negative number times a negative number is positive: (-4) \times (-3) = 12

Adding a negative number is the same as subtracting it: 10 + (-4) = 10 - 4 = 6

Worked example 2: Substituting a negative number

Work out the value of 4n + 7 when n = -3

  1. Replace n with (-3): 4 \times (-3) + 7

  2. Your turn. Multiply first. What is 4 \times (-3)?

    4 \times (-3) = -12

  3. Then add 7: -12 + 7 = -5
  4. Answer: -5

Subtracting a negative number

When b = -4, the expression 10 - b becomes 10 - (-4). Subtracting a negative number is the same as adding the positive number, so 10 - (-4) = 10 + 4 = 14

Only subtracting a negative number turns into adding. In -6 - 5 you are subtracting a positive number, so the answer goes further below zero: -6 - 5 = -11

Worked example 3: Two letters and two minus signs

Work out the value of p - 2q when p = 5 and q = -3

  1. Replace both letters: 5 - 2 \times (-3)

  2. Your turn. Multiply first. What is 2 \times (-3)?

    2 \times (-3) = -6, so the expression is 5 - (-6)

  3. Subtracting -6 is the same as adding 6: 5 + 6 = 11
  4. Answer: 11

Squares in a substitution

To square a number, multiply it by itself. A negative number squared is positive: (-4)^2 = (-4) \times (-4) = 16

In 3x^2 only the x is squared. Square first, then multiply by 3. When x = 2, 3x^2 = 3 \times 4 = 12, not 6^2 = 36

On a calculator, type a negative number in brackets before you square it: (-4)^2 gives 16, but -4^2 gives -16

Worked example 4: A square and a negative number

Work out the value of 2x^2 + 5x when x = -3

  1. Replace each x with (-3): 2 \times (-3)^2 + 5 \times (-3)

  2. Your turn. Square first. What is (-3)^2?

    (-3)^2 = 9, so the first term is 2 \times 9 = 18

  3. The second term is 5 \times (-3) = -15
  4. Add the two terms: 18 + (-15) = 18 - 15 = 3
  5. Answer: 3

Fraction bars

A fraction bar means divide the whole top by the whole bottom. Work out the top and the bottom first, then divide. For example, \frac{5 + 7}{4} = \frac{12}{4} = 3

On a calculator, use the fraction key, or put brackets round the whole top: (5 + 7) \div 4

Worked example 5: A formula with a fraction bar

Given y = \frac{3m + n}{2}, work out the value of y when m = 4 and n = -2

  1. Substitute: y = \frac{3 \times 4 + (-2)}{2}

  2. Your turn. Work out the top: 3 \times 4 + (-2)

    The top is 12 - 2 = 10

  3. Divide the whole top by 2: y = 10 \div 2 = 5
  4. Answer: y = 5

Worked example 6: Using a formula in context

The formula T = G - 0.006h gives the temperature, T°C, at a height of h metres above the ground, when the temperature on the ground is G°C. Work out the temperature at a height of 900 metres when the temperature on the ground is 4°C.

  1. Here G = 4 and h = 900: T = 4 - 0.006 \times 900

  2. Your turn. Multiply first. What is 0.006 \times 900?

    0.006 \times 900 = 5.4

  3. Subtract: T = 4 - 5.4 = -1.4
  4. Answer: -1.4°C

Common mistakes

  • Reading a number next to a letter as a two-digit number. A number next to a letter means multiply: when x = 7, 4x is 4 \times 7 = 28, not 47
  • Dropping the minus sign of a negative value. Put the value in brackets: when y = -4, 2y is 2 \times (-4), not 2 - 4
  • Saying “two minuses make a plus” every time. Only subtracting a negative number turns into adding: -8 - 2 is -10, not 10
  • Getting a negative answer when squaring a negative number. (-3)^2 = (-3) \times (-3) = 9, which is positive.
  • Multiplying before squaring. Square first: when x = 5, 2x^2 is 2 \times 25 = 50, not 10^2 = 100
  • Working from left to right. Multiply before you add or subtract: 1 + 3 \times (-2) is 1 - 6 = -5, not 4 \times (-2) = -8

Practice questions

More on this topic: Substitution worksheet with full solutions