Year 7 · Unit 3 Patterns, expressions and equations · Lesson 6 of 6

How do you find the number that makes both sides equal?

An equation says two things are equal. Keep the two sides balanced, undo what was done to the unknown, and check the answer by putting it back.

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 Unit 3, lesson 5, function machines. A machine multiplies by 4 and then adds 3. The output is 27. What was the input?
    Answer6. Undo the last step first: 27 − 3 = 24, then 24 ÷ 4 = 6.
  2. From Unit 3, lesson 4, collecting like terms. Simplify 5x + 2 + 3x.
    Answer8x + 2. 5x and 3x are like terms; the 2 stays on its own.
  3. From Unit 1, order of operations. What is (8 − 2) × 5?
    Answer30. The bracket first: 6, then 6 × 5.
Words used in this lesson
equation
a statement that two expressions are equal. For example: 3x + 4 = 19.
solve
find the value of the letter that makes the equation true. For example: solve 7x = 21.
solution
the value that makes both sides equal. For example: x = 3.
inverse
the step that undoes another. For example: divide undoes multiply.
balance
keep both sides equal. For example: take 4 from both sides.
check
put the answer back in the equation. For example: 3 × 5 + 4 = 19.

Two sides that are equal

At a spice market in Kumasi, a scale balances. On the left are 3 identical sacks and a 4 kg weight, and on the right is 19 kg. The sacks weigh the same as each other, but you do not know how much. Call it x kilograms. The scale tells you 3x + 4 = 19. An equation says that two expressions are equal, and solving it means finding the value of the letter that makes it true.

Predict first: 3 sacks and 4 kg balance 19 kg. What can you take from both sides and still keep the balance? What weight is one sack? Then step through.

3x + 4 = 19
0 of 3steps done
19the right-hand side

3x + 4 = 19: the left bar is 3 boxes of x and a box of 4, and it matches the bar of 19. What can you take from both?

Show me the steps
  1. 3x + 4 = 19: three boxes of x and a box of 4 are as long as the bar of 19.
  2. Take 4 from both sides: the left bar is three boxes of x, and the right bar is 15. So 3x = 15.
  3. Split each side into 3 equal parts: each box of x matches a part of 5. So x = 5.
  4. Check by substitution: 3 × 5 + 4 = 15 + 4 = 19. The solution makes the equation true.
  5. Equation 2, x + 6 = 10 (a = 1, b = 6): take 6 from both sides to get x = 4. There is no splitting step.
  6. Equation 3, 4x = 24 (a = 4, b = 0): there is nothing to take away, so split into 4 equal parts to get x = 6.

Whatever you do to one side, do to the other. Take the same from both sides, or divide both by the same number, and they stay equal. Use the opposite (inverse) steps, working in reverse order, until x is on its own.

An equation can be written either way round: 30 = 6k says the same as 6k = 30.

The same idea works for one-step equations: x + 5 = 12 needs one subtract; 6k = 30 needs one divide; p ÷ 5 = 3 needs one multiply. And you can always check by putting the answer back in.

Testing a guess

A solution is a value that makes both sides equal. You can find one by guessing and testing, and the bars show how close each guess is. Then the balancing method does it in one go.

3
4
19
4
16left side
19right side
too smallthe left side is

3x + 4 = 19 with x = 4: the left side is 3 × 4 + 4 = 16, which is 3 less than 19. Too small; try a larger x.

Show me the steps
  1. 3x + 4 = 19 with the guess x = 4: the left side is 3 × 4 + 4 = 16, and 16 is 3 less than 19. Too small.
  2. The guess x = 6: 3 × 6 + 4 = 22, which is 3 more than 19. Too big.
  3. The guess x = 5: 3 × 5 + 4 = 19. The sides are equal, so x = 5 is the solution.
  4. Change c to 20, so 3x + 4 = 20, and x = 5 gives 19 and x = 6 gives 22. Neither is 20, because (20 − 4) ÷ 3 is not a whole number. There is no whole-number solution.
  5. With a = 2, b = 9 and c = 9, the guess x = 0 gives 2 × 0 + 9 = 9. The sides are equal, so x = 0.

A common mistake: “3x + 4 = 19, so x = 19 ÷ 3 − 4.” That undoes the steps in the same order they were done. The 4 was added last, so it comes off first: 19 − 4 = 15, then 15 ÷ 3 = 5. Test it: 3 × 5 + 4 = 19.

Worked example: two steps and a check

Solve 4x + 9 = 41.

  1. Subtract 9 from both sides: 4x = 32.
  2. Divide both sides by 4: x = 8.
  3. Check: 4 × 8 + 9 = 32 + 9 = 41, which is the right side.

Your go. The first step is given. Solve 6x + 5 = 35 and check your answer.

  1. Subtract 5 from both sides: 6x = 30.
  2. Now finish the solving and then check by substitution yourself.
Check the answer

Divide both sides by 6: x = 5. Check: 6 × 5 + 5 = 30 + 5 = 35.

Check before you go on

1. To solve x + 9 = 25, what do you do to both sides?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Solve m + 7 = 10. What is m?

3. Solve 20 = 5k. What is k?

Level 2 · Building

4. Solve p ÷ 10 = 4. What is p?

5. Solve 5x + 13 = 28. What is x?

6. Which value of x makes 3x − 2 = 16 true?

Level 3 · Stretch

7. A tuk-tuk fare in rupees is 40 plus 15 for each kilometre, so 40 + 15k. The fare was 205 rupees. How many kilometres was the trip?

8. Kwame thinks of a number, doubles it and adds 15. The result is 49. What was his number?

Something to think about

Make up an equation of the form ax + b = c whose solution is x = 7. Check it by substitution. How many different equations like this are there?

One way to think about it

Pick any a and b, then c = 7a + b. For example 3x + 2 = 23, because 3 × 7 + 2 = 23. Or 5x + 1 = 36, because 5 × 7 + 1 = 36. There are as many as you like: every choice of a and b gives a new equation with the same solution.

Exit ticket

9. Tomas solves 3x + 6 = 24 and writes x = 24 ÷ 3 − 6 = 2. Which reply is right?