KS3 · Maths

Solving linear equations with unknowns on both sides

Here's a puzzle: 8x + 7 = 29 − 3x. There's x on both sides, so where do you start? Try the first moves below. Some work. One doesn't.

Maths · Equations

Pick a first move

Tap a first move for each equation and follow it to the end. Which moves get you somewhere, and which don't?

Equation → First move → Finish

6 routes to try.

On Two equations, lots of first moves. 2 branches to choose from.

Maths · Algebra

Solve it, line by line

Step forward and watch the balance. Every line is the one above it with the same thing done to both sides. Use the tabs for more equations.

GoalFind x in 5x + 1 = 2x + 10
1
5x + 1 = 2x + 10

Picture two bars of exactly the same length, one marked 5x + 1 and the other 2x + 10. (Or a balance with both pans level.) The equals sign promises that they match.

2
3
4
5
6
7

Step 1 of 7

Picture two bars of exactly the same length, one marked 5x + 1 and the other 2x + 10. (Or a balance with both pans level.) The equals sign promises that they match.

Maths · Your turn

A minus sign in the way

Solve x + 3 = 4x − 6. The first line is given. Pick the missing lines as you go.

  1. x + 3 = 4x − 6The −6 is in the way. Its zero-pair partner is +6, so add +6 to both sides.
  2. missing step
Which line is step 2?

Maths · Spot the slip

Where did it go wrong?

A student solves 5x + 3 = 6x + 2 and ends up with x = 1/3. Find the line where the working goes wrong.

A student's working — which line goes wrong?

Fractions on both sides

Problem

Solve (x + 5)/4 = (4x − 5)/6

Predict, then check

Have a go on paper first. Then commit to an answer.

A rectangle's opposite sides are 17 − 2x and 3x + 2. The other two sides are each 7. What is its perimeter?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Keep both sides balanced and you can start almost anywhere.

What you need to know

  • Solve equations with x on both sides by doing the same to both sides
  • Pick a first move that cancels something, and spot one that doesn't
  • Cope with negatives, fractions and answers that aren't whole numbers
  • Form an equation from equal lengths, and check any answer by substituting

The big picture

When x is on both sides of an equation, you keep the balance by doing the same thing to both sides until the unknowns are all on one side. There's no single right first move. What matters is that something cancels. Then you solve as usual and check by putting your answer back in.

Key points

1An equation says two expressions are equal, so whatever you do, you do to both sides.
2Cancel the x terms from one side with zero pairs (like +2x and −2x) until the unknown is on one side only. Then solve as usual.
3You can start with the x terms or the numbers: the order doesn't change the answer. A move that cancels nothing, like adding −3x to 8x + 7 = 29 − 3x, doesn't help.
4Subtracting from both sides only changes the matching term: 5x + 3 − 2 is 5x + 1, not 3x + 1.
5With fractions, multiply by the number that undoes them, or by a form of one to match denominators.
6Check by substituting your answer into the original equation: both sides must have the same value.

Worked example

Problem

Solve x + 13 = 3x − 5

⚠ Watch out

Taking a number off every term. When you subtract 2 from both sides of an equation with 5x + 3 in it, only the matching number changes: 5x + 3 − 2 is 5x + 1, not 3x + 1. The x term stays as it was.

🧠

Memory hook

Same to both sides. Cancel with zero pairs. Check by substituting.

✓

Check yourself

Solve 6x + 1 = 2x + 13 two different ways, then substitute to check. (Both give x = 3: 19 on each side.)

Flashcards

(14)
What does an equation tell you?
That two expressions are equal. Put the right value in for x and both sides give the same number: x = 5 in 3x − 1 = x + 9 gives 14 and 14.
What is the one rule for solving any equation?
Do the same to both sides. Then the two sides stay equal, like a balance that stays level.
What is a zero pair?
Two terms that add to zero, like +2x and −2x, or +4 and −4. Adding the partner to both sides makes the pair cancel.
In 8x + 7 = 29 − 3x, what do you add to cancel the −3x?
+3x, to both sides. Adding −3x doesn't cancel it: −3x + (−3x) makes −6x.
Do you have to start with the x terms, or the numbers?
Neither. There's no rule about what you do first. In x + 13 = 6x + 3, either order gives 10 = 5x and x = 2.
Do you have to remove the smaller x term first?
No. In 8x + 4 = 3x + 19 you can add −8x first (giving −15 = −5x and x = 3) or add −3x first. Both are fine.
What is 5x + 3 minus 2?
5x + 1. Subtracting only changes the matching term (the number), never the x term.
How do you solve x + 3 = 2x − 4?
Add +4 to both sides (a zero pair with the −4): x + 7 = 2x. Then x = 7.
5x + 8 = 7x + 5 leads to 2x = 3. What is x?
x = 3/2, which is 1.5. Not 3: 3 is the value of 2x, and you still have to divide by 2.
(1/5)x − 7 = 13 − 4x: what is a good first move?
Multiply both sides by 5, which undoes the × 1/5. That gives x − 35 = 65 − 20x.
How do you handle a fraction on both sides?
Multiply each side by a form of one, like 2/2 or 3/3, so the denominators match. Then the numerators must be equal. The lowest common multiple keeps the numbers small.
What can you say about the opposite sides of a rectangle?
They're equal in length. If they're written with x, set them equal to make an equation, solve it, then use x to find the actual lengths.
How do you check a solution?
Substitute it into the original equation. If both sides have the same value, it's right.
How can 5x + 1 = 2x + 10 be built from 3x = 9?
Add 1 to both sides to get 3x + 1 = 10, then add 2x to both sides. Solving it means undoing those moves in reverse.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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