Substitution
Substitution means replacing each pronumeral with its value and then working out the result. In NSW Year 8 maths the values are often negative, so each one goes in brackets before the order of operations is applied (NESA outcome MA4‑ALG‑C‑01).
You will find the value of expressions such as \(2x - 3y\) when \(x = 2\) and \(y = -2\), work with powers of negatives in \(2x^2 - 3y^2\), and evaluate fractions by working out the whole top and the whole bottom first. You will also test which expression gives a stated value. Freezer temperatures, bank balances in the red and a diver's depth all call for substituting negative numbers. It is part of Year 8 Algebra.
Theory
To substitute, replace each pronumeral with its number, put every negative value in brackets, then follow the order of operations. With \(x = 8\), \(y = 2\) and \(z = 6\): \(\dfrac{x + y}{2} + \dfrac{y - z}{4} = 5 - 1 = 4\). This Year 8 maths page shows how, with negative numbers and powers.
Write the hidden multiplication signs in: \(7ab = 7 \times a \times b\) and \(k^2 = k \times k\).
Put every negative value in brackets: if \(y = -7\), then \(2y = 2 \times (-7)\) and \(y^2 = (-7)^2 = 49\).
Calculate in order: brackets, then powers, then \(\times\) and \(\div\) from left to right, then \(+\) and \(-\) from left to right.
A fraction line groups the whole top and the whole bottom: work each out, then divide.
Powers of a negative. An even power gives a positive, an odd power a negative. A minus sign in front is not part of the power: \(-a^2\) with \(a = -5\) is \(-(25) = -25\).
Which expression takes a value? Substitute into each one in turn and compare.
One value of a, different expressions (a = −5):
| Expression | Working | Value |
|---|---|---|
| \(a^2\) | \((-5) \times (-5)\) | \(25\) |
| \(-a^2\) | \(-(25)\) | \(-25\) |
| \(2a^2\) | \(2 \times 25\) | \(50\) |
| \((2a)^2\) | \((-10) \times (-10)\) | \(100\) |
| \(a^3\) | \(25 \times (-5)\) | \(-125\) |
How to substitute
- Rewrite with \(\times\) signs and each value in brackets.
- Work out brackets and fraction tops and bottoms.
- Do powers, then \(\times\) and \(\div\), then \(+\) and \(-\).
| \(\) | \(=\) | \(\dfrac{8 + 2}{2} + \dfrac{2 - 6}{4}\) |
| \(\) | \(=\) | \(\dfrac{10}{2} - \dfrac{4}{4}\) |
| \(\) | \(=\) | \(5 - 1 = 4\) |
\(4\)
| \(4abc^2\) | \(=\) | \(12 \times (-2) \times 1\) |
| \(\) | \(=\) | \(-24\) |
| \(2a^2b^2c\) | \(=\) | \(2 \times 9 \times 4 \times (-1)\) |
| \(\) | \(=\) | \(-72\) |
| \(-24 - (-72)\) | \(=\) | \(48\) |
\(48\)
(ii) Find \(k(h - k)\) when \(h = 2\) and \(k = -5\).
| \(5 \times (-3) - 2 \times 4\) | \(=\) | \(-15 - 8\) |
| \(\) | \(=\) | \(-23\) |
| \((-5) \times (2 - (-5))\) | \(=\) | \((-5) \times 7\) |
| \(\) | \(=\) | \(-35\) |
(i) \(-23\)
(ii) \(-35\)
(ii) Find \(-x^2\) when \(x = -4\).
| \(3 \times (-4)^2 - 2^3\) | \(=\) | \(48 - 8 = 40\) |
| \(-((-4)^2)\) | \(=\) | \(-16\) |
(i) \(40\)
(ii) \(-16\)
| \(u^2v\) | \(=\) | \(4 \times 6 = 24\) |
| \(uv + 24\) | \(=\) | \(-12 + 24 = 12\) |
| \(\dfrac{v - u}{2}\) | \(=\) | \(\dfrac{8}{2} = 4\) |
Only \(uv + 24\) has the value \(12\).
| \(\dfrac{5 \times (-2) - (-6)}{4}\) | \(=\) | \(\dfrac{-4}{4}\) |
| \(\) | \(=\) | \(-1\) |
| \(ab\) | \(=\) | \((-2) \times (-6)\) |
| \(\) | \(=\) | \(12\) |
| \(-1 - 12\) | \(=\) | \(-13\) |
\(-13\)
Common pitfalls
Frequently asked questions
What does substitute mean?
Replace each pronumeral with a given number, then work out the value.
Why put negative numbers in brackets?
So powers and signs act on the whole number: \((-3)^2 = 9\).
What is the value of -a squared when a is negative?
Still negative: square first, then apply the minus. With \(a = -2\), \(-a^2 = -4\).
What order do I calculate in?
Brackets, powers, then multiply and divide, then add and subtract, each from left to right.
How do you substitute into a fraction?
Work out the whole top and the whole bottom, then divide.