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Linear & simultaneous equations · Lesson 1 of 3

Solving linear equations

An equation is a balance: do the same to both sides until the letter is alone, and clear fractions by multiplying every term.

15 minYou should already know: Expressions, formulae & change of subject
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A linear equation has one unknown letter, and the letter is never squared or cubed. For example, 5x−3=2x+95x - 3 = 2x + 9. Solving it means finding the value of xx that makes both sides equal.

An equation is a balance

The == sign says the two sides have the same value, like the two pans of a level balance. You can add, take away, multiply or divide, as long as you do the same to both sides. The balance stays level, and the answer doesn’t change.

5x − 32x + 9− 2x− 2xstill level: 3x − 3 = 9
Do the same to both sidesTaking 2x from both pans keeps the balance level
Keep it balancedChoose a move
5x − 32x + 9
5x − 3 = 2x + 9
The two sides weigh the same. Choose a move: it's done to both sides, so the scale stays level. Aim to get the x terms on one side and the numbers on the other.

Press the “left side only” button once to see what goes wrong when you break that rule.

Equations with fractions

Clear the fractions first. Find the LCM of the denominators: the lowest common multiple, the smallest number they all divide into. Multiply every term on both sides by it. The fractions disappear, and you’re back to an ordinary equation.

Worked example · NECO 2024

NECO 2024 · Paper 1 · Q19

Solve the equation 4x5−73=5x12\dfrac{4x}{5} - \dfrac73 = \dfrac{5x}{12}.

  1. Find the LCM of the denominators

    The denominators are 5, 3 and 12.

    Think first. What is the smallest number that 5, 3 and 12 all divide into?

  2. Multiply every term by 60

    60×4x5−60×73=60×5x1260 \times \frac{4x}{5} - 60 \times \frac73 = 60 \times \frac{5x}{12}48x−140=25x48x - 140 = 25x

  3. Collect the x terms

    48x−25x=14023x=140\begin{aligned} 48x - 25x &= 140 \\ 23x &= 140 \end{aligned}

    Think first. Take 25x25x from both sides. What's left?

  4. Divide

    x=14023=6223x = \frac{140}{23} = 6\tfrac{2}{23}

    The answer is D.

One fraction equals another: cross-multiply

When the equation is one fraction == one fraction, multiply both sides by both denominators. This is called cross-multiplying: top-left times bottom-right == top-right times bottom-left.

3m−n2m+n=144(3m−n)=1×(2m+n)\begin{aligned} \frac{3m - n}{2m + n} &= \frac14 \\ 4(3m - n) &= 1 \times (2m + n) \end{aligned}

Your turn

WAEC 2019 · Paper 1 · Q3

Solve y+12−2y−13=4\dfrac{y + 1}{2} - \dfrac{2y - 1}{3} = 4.

Worked solution (try it first)
  1. Multiply every term by 6, the LCM of 2 and 3: 3(y+1)−2(2y−1)=243(y + 1) - 2(2y - 1) = 24.
  2. Expand, taking care with the minus: 3y+3−4y+2=243y + 3 - 4y + 2 = 24, so −y+5=24-y + 5 = 24.
  3. Take 5 from both sides: −y=19-y = 19.
  4. Multiply by −1-1: y=−19y = -19, option B.

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