Substitutions
Substitution means replacing each pronumeral with a number and then working out the value of the expression. This NSW Year 7 maths subtopic (NESA outcome MA4‑ALG‑C‑01) uses positive and negative values as well as real formulas.
You will put the multiplication signs back in, place negative numbers in brackets and follow the order of operations: for \(m - n^2\) with \(m = 4\) and \(n = -2\), the value is \(4 - 4 = 0\). You will also use a formula such as \(A = \frac{1}{2}bh\) or a taxi fare \(F = 4 + 2.5d\), and give each answer with its units. This subtopic is part of Year 7 Algebra and Equations.
Every question with a fully worked solution.
- Substitutions - Video - Substitution Watch
Theory
To substitute, replace each pronumeral with its number, then work out the value using the order of operations. This Year 7 maths page shows how to substitute whole numbers, negative numbers and powers into expressions, how to use a formula, and how substitution checks an expansion.
A pronumeral can stand for different numbers. The expression \(x + 4\) is \(6\) when \(x = 2\), \(11\) when \(x = 7\) and \(24\) when \(x = 20\). Putting a number in place of a pronumeral is called substitution. The answer is the value of the expression.
Step 1. Write the multiplication signs back in: \(5a = 5 \times a\), \(ab = a \times b\), \(a^2 = a \times a\).
Step 2. Replace each pronumeral with its number. A negative number goes in brackets: \(3 \times (-2)\).
Step 3. Calculate in the order of operations: brackets, then powers, then \(\times\) and \(\div\), then \(+\) and \(-\). A fraction line groups the top, so work out the top first, then divide.
A formula is a rule that links pronumerals, such as \(P = 2l + 2w\). State the formula, substitute, calculate, then give the units.
Substituting makes a number pattern:
| \(n\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
|---|---|---|---|---|---|
| \(2n + 1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) |
How to substitute
- Write the multiplication signs back in.
- Replace each pronumeral with its number (negatives in brackets).
- Calculate: brackets, powers, \(\times\) and \(\div\), then \(+\) and \(-\).
- Write the answer, with units for a formula.
(ii) Complete the table for \(2x + 3\).
| \(x\) | \(1\) | \(2\) | \(5\) |
| \(2x + 3\) | ? | ? | ? |
| \(6 \times 3 - 5\) | \(=\) | \(18 - 5 = 13\) |
| \(2 \times 1 + 3\) | \(=\) | \(5\) |
| \(2 \times 2 + 3\) | \(=\) | \(7\) |
| \(2 \times 5 + 3\) | \(=\) | \(13\) |
(i) \(13\)
(ii) \(5, 7, 13\)
(ii) If \(x = 4y^2 + 5y\), evaluate \(x\) when \(y = 2\).
| \(3 \times 10 + 5\) | \(=\) | \(30 + 5 = 35\) |
| \(4 \times 2^2 + 5 \times 2\) | \(=\) | \(16 + 10 = 26\) |
(i) \(35\)
(ii) \(26\). Square first: \(2^2 = 4\), then \(4 \times 4 = 16\).
(ii) Find \(xyz\) when \(x = -2\), \(y = 5\) and \(z = 3\).
(iii) If \(p = 9\), \(q = -3\) and \(r = 4\), evaluate \(\dfrac{p - q}{r}\).
| \(3 \times 4 - 2 \times (-5)\) | \(=\) | \(12 + 10 = 22\) |
| \((-2) \times 5 \times 3\) | \(=\) | \(-30\) |
| \(\dfrac{9 - (-3)}{4}\) | \(=\) | \(\dfrac{12}{4} = 3\) |
(i) \(22\) (subtracting a negative is adding)
(ii) \(-30\)
(iii) \(3\) (work out the top first)
(ii) Find \(z^2 - 4z + 1\) when \(z = -5\).
Square first, with the negative in brackets: \((-2)^2 = 4\) and \((-5)^2 = 25\).
| \(3 \times (-2)^2\) | \(=\) | \(12\) |
| \((-5)^2 - 4 \times (-5) + 1\) | \(=\) | \(46\) |
(i) \(3 \times 4 = 12\). Only \(b\) is squared, not \(3b\).
(ii) \(25 + 20 + 1 = 46\)
(ii) The area of a rectangle is \(A = lb\). Find the area when \(l = 8\) m and \(b = 7\) m.
(iii) Use \(F = \dfrac{9}{5}C + 32\) to find \(F\) when \(C = 25\).
| \(150 + 12 \times 28\) | \(=\) | \(150 + 336 = 486\) |
| \(8 \times 7\) | \(=\) | \(56\) |
| \(\dfrac{9}{5} \times 25 + 32\) | \(=\) | \(45 + 32 = 77\) |
(i) \(\$486\)
(ii) \(56\) m²
(iii) \(F = 77\)
| \(C\) | \(=\) | \(2 \times \dfrac{22}{7} \times 14\) |
| \(\) | \(=\) | \(2 \times 22 \times 2\) |
| \(\) | \(=\) | \(88\) |
Divide first: \(14 \div 7 = 2\). The circumference is about \(88\) cm.
| \(5 \times (2 + 3)\) | \(=\) | \(25\) |
| \(5 \times 2 + 3\) | \(=\) | \(13\) |
| \(5 \times 2 + 15\) | \(=\) | \(25\) |
\(25 \ne 13\), so Lily is wrong: she did not multiply the \(3\) by \(5\). The correct expansion \(5x + 15\) also gives \(25\).
Common pitfalls
Frequently asked questions
What does substitution mean in algebra?
Replacing each pronumeral with a number and working out the value.
What is the value of 3x when x = 5?
\(3 \times 5 = 15\).
Why do negative numbers go in brackets?
So the sign stays with the number: \((-3)^2 = 9\).
What is a formula?
A rule that links pronumerals, such as \(P = 2l + 2w\).
How can substitution check an expansion?
Substitute the same number into both forms. If the values differ, the expansion is wrong.