Solving Linear Equations
Every question with a fully worked solution.
- Solving Linear Equations - Video - Solving Equations with variables on one side Watch
Theory
To solve an equation, find the value of the pronumeral that makes both sides equal. This Year 9 maths page shows how to undo the operations in reverse order, doing the same to both sides each time. It covers decimals, a negative coefficient, equations with a fraction, and checking the answer by substitution.
An equation says that two expressions are equal, such as \(3x + 5 = 20\). To solve it, find the value of the pronumeral that makes it true.
Think of an equation as a balance. Whatever you do to one side, do to the other. Then it stays balanced.
Each operation has an inverse that undoes it. Adding is undone by subtracting. Multiplying is undone by dividing.
Undo the operations in reverse order. In \(3x + 5\), the \(x\) was multiplied by \(3\) and then \(5\) was added. So subtract \(5\) first, then divide by \(3\).
The number in front of the pronumeral is its coefficient. In \(-2x\), the coefficient is \(-2\).
The inverse operations:
| Operation | Undo it by |
|---|---|
| \(+\) a number | \(-\) the same number |
| \(-\) a number | \(+\) the same number |
| \(\times\) a number | \(\div\) the same number |
| \(\div\) a number | \(\times\) the same number |
Clearing a fraction: multiply both sides by the denominator.
A whole expression over a denominator is cleared the same way:
How to solve an equation with the pronumeral on one side
- Undo any adding or subtracting outside the fraction or term, on both sides.
- Clear a fraction by multiplying both sides by its denominator.
- Divide both sides by the coefficient, keeping its sign.
- Check by substituting the answer into the original equation.
(i) \(4x + 7 = 31\)
(ii) \(6y - 5 = -2\).
Undo the adding or subtracting first. Then undo the multiplying.
(i)
| \(4x + 7 - 7\) | \(=\) | \(31 - 7\) |
| \(4x\) | \(=\) | \(24\) |
| \(x\) | \(=\) | \(\dfrac{24}{4}\) |
| \(x\) | \(=\) | \(\boldsymbol{6}\) |
Check: \(4 \times 6 + 7 = 31\) \(\checkmark\)
(ii)
| \(6y - 5 + 5\) | \(=\) | \(-2 + 5\) |
| \(6y\) | \(=\) | \(3\) |
| \(y\) | \(=\) | \(\dfrac{3}{6}\) |
| \(y\) | \(=\) | \(\boldsymbol{\dfrac{1}{2}}\) |
The answer does not have to be a whole number. \(y = \dfrac{1}{2}\) is the same as \(y = 0.5\).
The \(5\) is added on, so subtract \(5\) from both sides. Then divide by \(-2\).
| \(5 - 2m - 5\) | \(=\) | \(17 - 5\) |
| \(-2m\) | \(=\) | \(12\) |
| \(m\) | \(=\) | \(\dfrac{12}{-2}\) |
| \(m\) | \(=\) | \(\boldsymbol{-6}\) |
Check: \(5 - 2 \times (-6) = 5 + 12 = 17\) \(\checkmark\)
A positive divided by a negative is negative.
Multiply both sides by \(5\) to clear the fraction. Then divide by \(3\).
| \(\dfrac{3y}{5} \times 5\) | \(=\) | \(12 \times 5\) |
| \(3y\) | \(=\) | \(60\) |
| \(y\) | \(=\) | \(\dfrac{60}{3}\) |
| \(y\) | \(=\) | \(\boldsymbol{20}\) |
Check: \(\dfrac{3 \times 20}{5} = \dfrac{60}{5} = 12\) \(\checkmark\)
The \(+\,2\) is outside the fraction, so deal with it first. Then clear the fraction.
| \(\dfrac{3a - 1}{4} + 2 - 2\) | \(=\) | \(7 - 2\) |
| \(\dfrac{3a - 1}{4}\) | \(=\) | \(5\) |
| \(\dfrac{3a - 1}{4} \times 4\) | \(=\) | \(5 \times 4\) |
| \(3a - 1\) | \(=\) | \(20\) |
| \(3a\) | \(=\) | \(21\) |
| \(a\) | \(=\) | \(\boldsymbol{7}\) |
Check: \(\dfrac{3 \times 7 - 1}{4} + 2 = \dfrac{20}{4} + 2 = 7\) \(\checkmark\)
Common pitfalls
Frequently asked questions
How do you solve a two-step equation?
Undo the adding or subtracting first, then undo the multiplying or dividing. Do the same to both sides each time. For \(4x + 7 = 31\), subtract \(7\) to get \(4x = 24\), then divide by \(4\) to get \(x = 6\).
What is an inverse operation?
It is the operation that undoes another one. Subtracting undoes adding, and dividing undoes multiplying.
How do you solve an equation with a fraction?
Multiply both sides by the denominator to clear the fraction. Then solve the equation that is left. If a number is added outside the fraction, take it away first.
What do you do when the pronumeral has a negative in front?
Divide both sides by the negative number. For \(-2m = 12\), divide by \(-2\) to get \(m = -6\).
How do you check the answer to an equation?
Substitute the answer into the original equation. Work out each side. If they are equal, the answer is correct.