Year 8 · Unit 4 Expressions, equations and inequalities · Lesson 3 of 6

What do you do first when there is a bracket?

Keep both sides balanced, undo in reverse order, and choose between dividing first and expanding first when a bracket is in the way.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7, Unit 3, lesson 6, solving simple equations. Solve 5x + 6 = 31.
    Answerx = 5. 31 − 6 = 25, then 25 ÷ 5 = 5.
  2. From Lesson 1 of this unit. Expand 6(x + 2).
    Answer6x + 12. The 6 multiplies the x and the 2.
  3. From Unit 3 (integers), lesson 3. Work out 4 − 10.
    Answer−6. Ten steps down from 4 passes zero and ends 6 below it.
Words used in this lesson
equation
a statement that two expressions are equal. For example: 3(x + 4) = 27.
solve
find the value that makes the equation true. For example: x = 5.
both sides
the left and the right of the equals sign. For example: divide both sides by 3.
inverse
the step that undoes another. For example: subtract undoes add.
check
put the answer back in the equation. For example: 3 × (5 + 4) = 27.

Which move comes first?

Three friends at a stall in Sharjah each buy the same notebook, x dirhams, and a ruler for 4 dirhams. They pay 27 dirhams in all, so 3(x + 4) = 27. A bracket means the same thing is bought three times. What did one notebook cost?

Predict first: what can you do to both sides of 3(x + 4) = 27 to make it simpler? Is there more than one way? Try each route with the buttons. The boxes are drawn to scale, but find x by balancing, not by measuring.

3(x + 4) = 27
0 of 3steps done
27the right side

3(x + 4) = 27: 3 groups, each an x box and a box of 4, are as long as the bar of 27. What can you do to both sides?

Show me the steps
  1. 3(x + 4) = 27: three groups, each an x box and a box of 4, are as long as the bar of 27.
  2. Divide first: divide both sides by 3. One group is left and the bar is 9, so x + 4 = 9.
  3. Subtract 4 from both sides: x = 5.
  4. Check by substitution: 3 × (5 + 4) = 3 × 9 = 27, which matches.
  5. Expand first: 3(x + 4) becomes 3x + 12 = 27. Subtract 12: 3x = 15. Divide by 3: x = 5. The same answer by a longer road.
  6. Equation 2, 4(x + 3) = 20: divide first gives x + 3 = 5, so x = 2. Expanding first gives 4x + 12 = 20, 4x = 8, x = 2.

Whatever you do to one side, do to the other. Work backwards through what was done to x, until x stands alone. Then check by putting the answer back in.

With a bracket there are two routes, and both reach the same x. Divide first when the number outside divides the right side exactly; otherwise expanding first keeps the working in whole numbers for longer.

Two slips to watch for. 2(x + 5) = 18 does not give 2x + 5 = 18: the 2 multiplies the 5 as well, so it is 2x + 10 = 18. And in −2x = 12 the x is multiplied by −2, so divide by −2: x = −6, not 6.

Worked examples

Solve 5(x − 2) = 35 by dividing first.

  1. Divide both sides by 5: x − 2 = 7.
  2. Add 2 to both sides: x = 9.
  3. Check: 5 × (9 − 2) = 5 × 7 = 35.

Solve 3(2x + 1) = 27 by expanding first.

  1. Expand: 6x + 3 = 27.
  2. Subtract 3 from both sides: 6x = 24.
  3. Divide both sides by 6: x = 4. Check: 3 × 9 = 27.

Solve −3x + 4 = 19.

  1. Subtract 4 from both sides: −3x = 15.
  2. Divide both sides by −3: x = −5.
  3. Check: (−3) × (−5) + 4 = 15 + 4 = 19.

Your go. The first step is given. Solve 2(x + 8) = 10.

  1. Divide both sides by 2: x + 8 = 5.
  2. Finish the solving, then check by substitution.
Check the answer

Subtract 8: x = −3. Check: 2 × (−3 + 8) = 2 × 5 = 10.

Check before you go on

1. Dividing both sides of 5(x + 2) = 40 by 5 gives which equation?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Solve x − 7 = −3. What is x?

3. Solve −4x = 28. What is x?

Level 2 · Building

4. Solve 3x − 8 = 13. What is x?

5. Solve 4(x + 5) = 52. What is x?

6. Which value of x makes 2(3x − 4) = 10 true?

Level 3 · Stretch

7. Five friends in Colombo each buy a T-shirt for 600 rupees and each use a voucher worth d rupees off. They pay 2750 rupees in all. What is d?

8. Solve 5 − 2x = 17. What is x?

Something to think about

Solve 5(2x − 3) = 35 twice: dividing first, then expanding first. Do you get the same x? Which route did you find shorter, and why?

One way to think about it

Both give x = 5. Dividing first: 2x − 3 = 7, 2x = 10, x = 5. Expanding first: 10x − 15 = 35, 10x = 50, x = 5. Dividing first is shorter here because 35 ÷ 5 is a whole number. If the right side were 36, dividing first gives 2x − 3 = 7.2 straight away; expanding first keeps whole numbers until 10x = 51, so x = 5.1 either way.

Exit ticket

9. Kofi solves 4(x + 6) = 36 and writes: 4x + 6 = 36, 4x = 30, x = 7.5. Which reply is right?