Substitutions
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.