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

What if x is on both sides?

Take the same number of x's from both sides so that x is on one side only, then solve as before. A comparison of two plans shows why the answer is where the two sides are equal.

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 4, collecting like terms. Simplify 7x − 2x + 3.
    Answer5x + 3. 7x and −2x are like terms.
  2. From Lesson 3 of this unit. Solve 3(x + 2) = 18.
    Answerx = 4. Divide by 3 to get x + 2 = 6, then subtract 2.
  3. From Unit 3 (integers), lesson 3. Work out −9 + 14.
    Answer5. From −9, 9 steps reach zero and 5 more go beyond.
Words used in this lesson
both sides
the left and the right of the equals sign. For example: take 3x from both sides.
collect
bring terms of the same kind together. For example: collect the x terms.
equal
the same in value. For example: 18 = 18.
solution
the value that makes both sides equal. For example: x = 2.
fee
a fixed charge that does not change. For example: 36 taka a month.

When do two plans cost the same?

A phone company in Dhaka offers two data plans. Plan A has a 36 taka fee a month and costs 4 taka for each gigabyte. Plan B has no fee and costs 7 taka for each gigabyte. For 5 GB, which is cheaper? For 20 GB? Is there an amount where they cost exactly the same?

Predict first: which plan do you think is cheaper for 5 GB, and which for 20 GB? Then use the figure, pair 7, to compare the plans at different amounts of data.

A: 12 + 3x, B: 4 + 7x
0
12plan A costs, in taka
4plan B costs, in taka
plan A costs 8 taka morethe difference

At x = 0 plan A costs 12 and plan B costs 4 taka, so plan A costs 8 taka more. Move x to find where they meet.

Show me the steps
  1. Pair 1, plan A is 12 + 3x and plan B is 4 + 7x. At x = 0, A costs 12 and B costs 4, so plan A costs 8 taka more.
  2. Move x to 1: A is 15 and B is 11, so plan A costs 4 taka more: still dearer, but by less.
  3. At x = 2: A is 18 and B is 18. The plans cost the same. This x is the solution of 3x + 12 = 7x + 4.
  4. At x = 4: A is 24 and B is 32, so plan B costs 8 taka more: the plan that started cheaper has become dearer.
  5. Pair 7, plan A is 36 + 4x and plan B is 7x, the plans from the opening. Which x makes them cost the same?

The solution is where the two sides are equal. To find it without trying every value, put x on one side only: take the same number of x's from both sides, then take the same number from both sides, then divide.

Take away the smaller number of x's when you can, so the x term that is left stays positive; if x is taken away on the left, add it. Then check by substituting into both sides.

Balancing it

The same two bars, one step at a time. Notice that each step keeps the bars the same length. The boxes are drawn to scale, but find x by balancing, not by measuring.

3x + 12 = 7x + 4
0 of 4steps done
18the value of both sides at the solution

3x + 12 = 7x + 4: the two bars are equal in length, with x on both sides. What can you take from both?

Show me the steps
  1. 3x + 12 = 7x + 4: the two bars are equal.
  2. Take 3 x-boxes from both sides, so x is on the right only: 12 = 4x + 4.
  3. Take 4 from both sides: 8 = 4x.
  4. Divide both sides by 4: x = 2.
  5. Check by substitution: 3 × 2 + 12 = 18 and 7 × 2 + 4 = 18. Equal.
  6. Equation 6, 2x + 17 = 3x + 8: take 2 x-boxes, then take 8, and 9 = x is already the answer, so there is no dividing step.

Take it from both sides. In 8x + 3 = 5x + 24, subtracting 5x from the right only would give 8x + 3 = 24: the bars are no longer equal. Whatever leaves one side must leave the other.

Worked examples

Back to the data plans: when is 4x + 36 equal to 7x?

  1. Subtract 4x from both sides: 36 = 3x.
  2. Divide both sides by 3: x = 12 gigabytes.
  3. Check: 36 + 4 × 12 = 84 and 7 × 12 = 84 taka. Below 12 GB the plan with no fee is cheaper; above it, the plan with the fee is.

Solve 5x + 3 = 3x + 11.

  1. Subtract 3x from both sides: 2x + 3 = 11.
  2. Subtract 3 from both sides: 2x = 8.
  3. Divide both sides by 2: x = 4. Check: 5 × 4 + 3 = 23 and 3 × 4 + 11 = 23.

Solve 4(x − 1) = 2x + 10.

  1. Expand: 4x − 4 = 2x + 10.
  2. Subtract 2x: 2x − 4 = 10. Add 4: 2x = 14.
  3. Divide by 2: x = 7. Check: 4 × 6 = 24 and 2 × 7 + 10 = 24.

Solve 3 − 2x = x + 12. Here x is taken away on the left, so add 2x to both sides.

  1. Add 2x to both sides: 3 = 3x + 12.
  2. Subtract 12: −9 = 3x. Divide by 3: x = −3.
  3. Check: 3 − 2 × (−3) = 9 and −3 + 12 = 9.

Your go. The first step is given. Solve 7x − 4 = 3x + 20.

  1. Subtract 3x from both sides: 4x − 4 = 20.
  2. Finish the solving, then check by substitution into both sides.
Check the answer

Add 4: 4x = 24, so x = 6. Check: 7 × 6 − 4 = 38 and 3 × 6 + 20 = 38.

Check before you go on

1. Subtract 2x from both sides of 6x + 5 = 2x + 21. Which equation do you get?

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 7x = 3x + 20. What is x?

3. Solve 5x + 8 = 2x + 26. What is x?

Level 2 · Building

4. Solve 9x − 7 = 4x + 28. What is x?

5. Solve 3(x + 4) = x + 20. What is x?

6. Which equation has the solution x = 3?

Level 3 · Stretch

7. Shop A rents a bike for 80 pesos plus 15 pesos per hour. Shop B charges 20 pesos plus 25 pesos per hour. After how many hours do both cost the same?

8. Solve 2(3x − 1) = 4(x + 5) − 6. What is x?

Something to think about

Try to solve 2x + 3 = 2x + 8. What happens when you take 2x from both sides? What does that tell you about the two plans it describes?

One way to think about it

Taking 2x from both sides leaves 3 = 8, which is never true. No value of x works: the right side is always 5 more than the left, so the two plans never cost the same. An equation can have no solution.

Exit ticket

9. Ana solves 8x − 5 = 3x + 10 and writes: 8x + 3x − 5 = 10, so 11x = 15. Which reply is right?