GCSE · Maths · AQA · Spec 8300 · Foundation

Linear equations with unknown on both sides

Try x = 3 in 5x − 3 and 2x + 7: you get 12 and 13. Try x = 4: 17 and 15. So where exactly are they equal?

Where are the two sides equal?

01234-4281420xvalue of each side(2, 7)

x: 2. value of each side: 7

Drag the point to where the two lines cross

The other line is the right-hand side, y = 2x + 7: Your point starts at x = 2, where the left side is 7 but the right side is 11. Drag to the right and the left side catches up, because it grows by 5 for every 1 added to x while the right side grows by only 2. The lines meet between x = 3 and x = 4, a little past 3.3, where both sides are about 13.7. That x-value is the solution.

Watch out: The answer is the x-value of the crossing point, not its height. The height (about 13.7) is what both sides are worth there.

Maths · Algebra

Now find it exactly

Same equation as the graph. Every line does one thing to both sides. Step through and watch the x-terms gather on one side.

GoalSolve 5x − 3 = 2x + 7
1
5x − 3 = 2x + 7

x is on both sides. First job: get all the x-terms together on one side.

2
3
4
5

Step 1 of 5

x is on both sides. First job: get all the x-terms together on one side.

Watch out: Say the operation out loud: 'subtract 2x from both sides'. If you can't name what you did to both sides, the line is probably wrong.

Your turn

Fill in the missing steps

Solve 4(2x − 1) = 3(x + 7)

  1. 4(2x − 1) = 3(x + 7)
  2. missing step
Which line is step 2?

Spot the slip

Where does this answer go wrong?

Solve 2(3x + 1) = 4x − 8

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two expressions, one question: for which x are they equal?

What you need to know

  • Solving an equation means finding the value of x that makes both sides equal.
  • Whatever you do to one side, do to the other. That keeps every line of your working true.
  • Expand brackets first, multiplying every term inside the bracket.
  • On a graph of both sides, the solution is the x-value where the two lines cross. A graph gives it only approximately.

The big picture

An equation with the unknown on both sides asks which value of x makes the two sides equal. Draw each side as a line and the solution is the x-value where they cross, which gives an approximate answer. For the exact answer, do the same operation to both sides: expand any brackets, collect the x-terms on one side and the numbers on the other, then divide.

Key points

1Collect the x-terms on the side that has more x's: subtract the smaller x-term from both sides, so the x-term left is positive.
2Then clear the number beside the x-term with the inverse operation, and divide both sides by the number in front of x.
3x can finish on the right-hand side: 20 = 4x means x = 5.
4Check by substituting your answer into both sides. They must give the same value.
5From a graph, read the x-coordinate of the crossing point, not the y-coordinate. Its accuracy is limited by the scale.

Worked example

Problem

Solve 9 − 2x = 3x − 11.

⚠ Watch out

Treating solving as 'move it over and change the sign'. That shortcut is where sign slips come from. Name the operation instead: 'subtract 4x from both sides' can only give 6x − 4x, never 6x + 4x.

🧠

Memory hook

Two lines, one crossing. The graph shows roughly where the sides are equal; the balance (same thing to both sides) lands on it exactly.

✓

Check yourself

Solve 6x − 5 = 2x + 9. Then put your answer into both sides. If the two sides don't come out equal, go back and find the line where your working went wrong.

Flashcards

(7)
What does it mean to solve an equation?
Find the value of the unknown that makes both sides equal.
What keeps an equation true while you solve it?
Doing the same operation to both sides, like keeping a balance level.
x is on both sides. What's your first move?
Collect the x-terms on one side: add or subtract an x-term on both sides. Pick the side with more x's so the x-term left is positive.
How do you deal with a bracket like 4(x − 3)?
Expand it first. Multiply every term inside: 4(x − 3) = 4x − 12.
How does a graph show the solution of an equation with x on both sides?
Draw y = left side and y = right side. The solution is the x-coordinate where the lines cross.
Why is a graph's answer only approximate?
You can only read the crossing point as accurately as the scale allows. Algebra gives the exact value.
How do you check your answer?
Substitute it into both sides. If both sides give the same value, it's right.

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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