An equation is a balance: do the same to both sides and it stays true. Solve equations with brackets and unknowns on both sides, form your own from a problem in words, and (Extended) clear fractions first.
Whatever you do to one side of an equation, do to the other. Expand brackets, collect the unknown on one side and the numbers on the other, then divide. Check by substituting your answer back into the original equation.
Section 01
Solving on a balance.
In the lab pick an equation and press Next step one move at a time. Each step takes the same bags or blocks off both pans, so the balance stays level. Before each press, say what you would do next, then check against the lab.
Worked example Two steps
Solve \(3x + 4 = 10\).
Subtract 4 from both sides: \(3x = 6\).
Divide both sides by 3.
Answer: \(x = 2\)
Worked example Brackets on both sides
Solve \(4(2x - 3) = 2(x + 9)\).
Expand: \(8x - 12 = 2x + 18\).
Subtract \(2x\) from both sides: \(6x - 12 = 18\).
Add \(2x\) to both sides so the \(x\) term is positive: \(5 = 5x + 21\).
Subtract 21: \(-16 = 5x\).
Divide by 5: \(x = -\dfrac{16}{5}\).
Answer: \(x = -3.2\)
Your turn
Solve \(8 - 3(x - 2) = 2(x + 4)\).
Expand: \(8 - 3x + 6 = 2x + 8\), so \(14 - 3x = 2x + 8\). Add \(3x\) and subtract 8: \(6 = 5x\), so \(x = 1.2\).
Examiner trap
In \(8 - 3(x - 2)\), the \(-3\) multiplies the \(-2\) as well: \(-3 \times (-2) = +6\). Writing \(8 - 3x - 6\) is the most common slip on this type.
Section 02
Forming an equation.
Choose a letter for the unknown, write each quantity in terms of it, then use a fact you know (angles in a triangle add to 180°, a perimeter, a total) to make an equation.
Worked example Angles in a triangle
The angles of a triangle are \(x\)°, \((2x + 10)\)° and \((3x - 40)\)°. Find the size of the largest angle.
Angles in a triangle add to 180°: \(x + 2x + 10 + 3x - 40 = 180\).
Collect: \(6x - 30 = 180\), so \(6x = 210\) and \(x = 35\).
The angles are \(35\)°, \(2(35) + 10 = 80\)° and \(3(35) - 40 = 65\)°.
Answer: The largest angle is 80°.
Worked example A perimeter
A rectangle is 3 cm longer than twice its width. Its perimeter is 42 cm. Find its length.
Let the width be \(w\) cm; the length is \((2w + 3)\) cm.
The sum of three consecutive whole numbers is 87. Find the numbers.
\(n + (n + 1) + (n + 2) = 87\), so \(3n + 3 = 87\), \(3n = 84\), \(n = 28\). The numbers are 28, 29 and 30.
Examiner trap
Finding \(x\) is often not the end. Read the question again: it may ask for an angle, a length or an age built from \(x\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Solve \(3x + 4 = 10\).
Quick check+10 XP
Solve \(5x = 2x + 12\).
Quick check+10 XP
Solve \(\dfrac{x}{3} - 2 = 5\).
Quick check+10 XP
I think of a number \(n\), double it and subtract 7. The answer is 15. Which equation is correct?
Section 03
Fractional equations (Extended).
Clear the fractions first. Multiply every term on both sides by the lowest common denominator; when the denominators contain \(x\), cross-multiply. Put brackets round each numerator.