From one-step equations up to forming and solving from words.
6 lessons · 35 questions (A–G) · 20 calculation + 15 word problems · no time pressure
Solving a linear equation means finding the value of the letter that makes both sides balance, by doing the same thing to each side until the letter is on its own. Pupils most often slip by changing one side but not the other, mishandling a negative when moving a term across, or expanding a bracket wrongly before solving.
At Improve Tuition, a qualified teacher uses the balance idea to make every step make sense, in small worked steps — with read-aloud, comfort spacing and a reading tint for pupils who take in maths more easily that way.
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Lesson 1
One-step equations
Aim: Solve an equation in one step using the inverse operation.
An equation is a balance: whatever you do to one side, do to the other. To get x on its own, undo what is done to it with the inverse — add undoes subtract, multiply undoes divide.
The scales balance because the sides are equal. Take 7 off each side to leave x = 5.
Worked example
Solve x + 7 = 12.
We want x on its own. At the moment 7 is being added to it.
To undo ‘+ 7’, do the opposite — subtract 7. Do it to both sides so the scales stay balanced.
x + 7 − 7 = 12 − 7.
On the left the +7 and −7 cancel, leaving x = 5.
Check: 5 + 7 = 12. ✓
Watch out
Do the same to both sides — that keeps the balance.
Try one
Solve x − 4 = 9.
Show answer
x − 4 = 9
What is happening to x? 4 has been subtracted.
The opposite of − 4 is + 4, so add 4 to both sides:
Aim: Solve equations with the unknown on both sides.
Collect the x terms on one side and the numbers on the other. Subtract the smaller x term from both sides so x stays positive, then finish as a two-step equation.
x’s sit on both pans. Take 2x off each side, leaving 3x = 12, so x = 4.
Worked example
Solve 5x = 2x + 12.
x appears on both sides, so first gather all the x’s on one side.
Take 2x off both sides: 5x − 2x = 12, which leaves 3x = 12.
Now undo the ‘× 3’ by dividing both sides by 3.
x = 12 ÷ 3 = 4.
Check: 5 × 4 = 20, and 2 × 4 + 12 = 20. ✓
Watch out
Move the smaller x term to avoid a negative coefficient.
Try one
Solve 7x − 2 = 4x + 13.
Show answer
7x − 2 = 4x + 13
x is on both sides. Move the smaller x term first.
31.I think of a number, double it and add 5. The result equals the number plus 12. Find the number.
Answer:
Show working
Form the equation: 2x + 5 = x + 12.
Subtract x from both sides: x + 5 = 12.
Subtract 5 from both sides: x = 7.
32.Sam is 4 years older than Tom. Together they are 26. How old is Tom?
Answer:
Show working
Let Tom be t, so Sam is t + 4.
Form the equation: t + (t + 4) = 26.
Collect like terms: 2t + 4 = 26.
Subtract 4 from both sides: 2t = 22.
Divide both sides by 2: t = 11.
33.A father is 3 times as old as his son. Their ages add to 48. How old is the son?
Answer:
Show working
Let the son be s, so the father is 3s.
Form the equation: s + 3s = 48.
Collect like terms: 4s = 48.
Divide both sides by 4: s = 12.
34.I think of a number, multiply by 4 and add 3. This equals the number multiplied by 6 then subtract 7. Find the number.
Answer:
Show working
Form the equation: 4x + 3 = 6x − 7.
Subtract 4x from both sides: 3 = 2x − 7.
Add 7 to both sides: 10 = 2x.
Divide both sides by 2: x = 5.
35.A rectangle's perimeter is 40 cm. Its length is 4 cm more than its width. Find the width.
Answer:
Show working
Let the width be w, so the length is w + 4.
Perimeter: 2(w + w + 4) = 40.
Collect like terms: 2(2w + 4) = 40.
Divide both sides by 2: 2w + 4 = 20.
Subtract 4 from both sides: 2w = 16.
Divide both sides by 2: w = 8 cm.
0 of 35 answered
Guide
How do you solve a linear equation?
A linear equation has a letter to find and an equals sign, such as 3x + 4 = 19. Solving it means working out the value of the letter that makes both sides equal — here, x = 5.
The method is to keep the equation balanced: whatever you do to one side, do to the other. Undo it step by step — first remove anything added or subtracted, then anything multiplying or dividing — until the letter stands alone. With brackets, expand first; with the letter on both sides, collect it onto one side.
Common mistakes. Changing one side but not the other, getting a sign wrong when moving a term across, expanding a bracket incorrectly, or dividing only part of a side.
Common questionsWhat does it mean to solve an equation?
It means finding the value of the unknown letter that makes both sides of the equation equal. For 2x = 10, the solution is x = 5.
What is the balance method?
You treat the equation like a balanced scale: whatever you do to one side, you do the same to the other, so it stays equal while you work towards the letter on its own.
How do you solve an equation with brackets?
Expand the bracket first by multiplying everything inside by the term outside, then solve the resulting equation as normal.
How do you solve an equation with the letter on both sides?
Collect the letter terms onto one side and the numbers onto the other by adding or subtracting the same thing from both sides, then solve.
How can you check the solution to an equation?
Substitute your answer back into the original equation. If both sides come out equal, the solution is correct.
Stuck on this topic?
A teacher can find the exact gap
Practising linear equations on your own is a strong start. If the same marks keep slipping, a qualified teacher can pinpoint the precise gap and fix it. Improve Tuition offers one-to-one maths tuition with our maths tutors in Batley and online — the first assessment is free.