Substitution
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.