Chapter 6 of 8

All 8 chapters
  1. 1Letters for unknown numbers, readunderstood
  2. 2Putting numbers in, readunderstood
  3. 3Collecting like terms, readunderstood
  4. 4Brackets, readunderstood
  5. 5Solving equations in one step, readunderstood
  6. 6Solving equations in two steps, readunderstood
  7. 7Writing your own equations, readunderstood
  8. 8Sequences and the nth term, readunderstood

M3.8.6 Chapter 6

Solving equations in two steps

A balance is level. The left pan holds 3 bags of x stones and 5 loose stones. The right pan holds 20 loose stones. How many stones are in each bag?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Solve 3x + 5 = 20
  • Solve equations with brackets, like 2(x + 3) = 14
  • Solve 5x − 2 = 3x + 6, with x on both sides

In the last chapter each equation took one step. Now the bag is multiplied and a number is added too. The equation 3x + 5 = 20 needs two steps. It is a two-step equation.

Draw it first

Two steps on the balance

  1. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xxx3x + 520

    1 Three bags and 5 stones balance 20 stones.

  2. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xxx3x15

    2 Take 5 stones from each side. Three bags balance 15 stones.

  3. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xx53 groups of 5

    3 Share each side into 3 equal groups. The 15 stones make 3 groups of 5. One bag balances 5 stones.

  4. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xx53 groups of 5x = 5

    4 So x = 5.

The order matters. In 3x + 5, x was first multiplied by 3, and then 5 was added. We undo the steps backwards. Take away 5 first, then divide by 3.

A number machine. The number x = 5 goes in. The first step is × 3 and gives 15. The second step is + 5 and gives 20.5x× 315+ 520
Building 3x + 5: multiply by 3, then add 5.
A number machine that runs backwards. The number 20 goes in. The first step is − 5 and gives 15. The second step is ÷ 3 and gives 5. The letter x is written over the last box.20− 515÷ 35x
Solving it: start at 20, take away 5, then divide by 3.

Worked example

Solve 3x + 5 = 20.

  1. Take 5 from both sides: 3x = 20 − 5 = 15.
  2. Divide both sides by 3: x = 15 ÷ 3 = 5.
  3. Check: put 5 back. 3 × 5 + 5 = 15 + 5 = 20. It is true.

Answer: x = 5

Some equations have brackets. In 2(x + 3) = 14, the 2 multiplies both x and 3. One way is to expand the bracket first.

A rectangle 2 high. Its width is x + 3, with x on the left part and 3 on the right part. The left part has the area 2x and the right part has the area 6.x32x + 32x6
2(x + 3) = 2x + 6. The whole rectangle has the area 14.

Worked example

Solve 2(x + 3) = 14.

  1. Expand the bracket: 2(x + 3) = 2x + 6, so 2x + 6 = 14.
  2. Take 6 from both sides: 2x = 14 − 6 = 8.
  3. Divide both sides by 2: x = 8 ÷ 2 = 4.
  4. Check: put 4 back. 2 × (4 + 3) = 2 × 7 = 14. It is true.

Answer: x = 4

There is another way. Divide both sides by 2 first. This gives x + 3 = 7, and then x = 4. Both ways give the same answer.

See it three ways

With things

Picture a see-saw. On one side are 2 bags of stones and 6 loose stones. On the other side are 14 loose stones.

As a picture

A level balance. The left side has 2 bags of x stones and 6 loose stones, with the words 2x + 6. The right side has 14 loose stones, with the words 14.xx2x + 614

In symbols

2(x + 3) = 14 becomes 2x + 6 = 14. Then 2x = 8 and x = 4.

Sometimes x is on both sides. First take the same number of bags from both sides. This leaves the bags on one side only.

Draw it first

Bags on both sides

  1. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xxxxx5xxxx3x + 8

    1 Five bags balance 3 bags and 8 stones. This is 5x = 3x + 8.

  2. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xx2x8

    2 Take 3 bags from each side. Two bags balance 8 stones.

  3. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xx42 groups of 4

    3 Share each side into 2 equal groups. The 8 stones make 2 groups of 4. One bag balances 4 stones.

  4. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xx42 groups of 4x = 4

    4 So x = 4.

Worked example

Solve 5x − 2 = 3x + 6.

  1. Add 2 to both sides: 5x = 3x + 6 + 2 = 3x + 8.
  2. Take 3x from both sides: 5x − 3x = 8, so 2x = 8.
  3. Divide both sides by 2: x = 8 ÷ 2 = 4.
  4. Check the left side: 5 × 4 − 2 = 20 − 2 = 18.
  5. Check the right side: 3 × 4 + 6 = 12 + 6 = 18. Both sides match.

Answer: x = 4

Try it

Two steps on a stick balance

You need: a ruler or flat stick, a pencil to rest it on, 6 cups that are the same, 50 small stones or washers, all the same size.

  1. Ask a friend to put the same number of stones in each of 3 cups. Use 2 to 6 stones. Cover them. Rest the stick on the pencil.
  2. Put the 3 full cups and 5 loose stones on the left. Put 3 empty cups on the right, so the cups cancel. Add loose stones to the right until the stick is level.
  3. Take 5 stones from each side. Is the stick still level?
  4. Share the stones on the right into 3 equal groups. One group is the number of stones in one cup.

What you will see: First you take away the loose number. Then you share. The empty cups cancel the full ones, and the stick stays level at every step.

If you cannot try it, this is what happens.
A level balance. The left side has 3 bags of x stones, with the words 3x. The right side has 15 loose stones, with the words 15.xxx3x15

A wrong idea many people have

In 3x + 5 = 20, divide only the 3x by 3, and leave the rest.

Why it feels right: The 3 is stuck to the x, so it seems the only thing to divide.

The test that breaks it: Try it. You get x + 5 = 20, so x = 15. Put 15 back: 3 × 15 + 5 = 50, not 20.

What is true: Take away the 5 first. Or divide every term on both sides by 3.

Key words

two-step equation
an equation that needs two steps to solve, such as 3x + 5 = 20.

Back to your guess

The answer is A: x = 5. Take 5 stones from each side, then share the 15 stones into 3 equal groups.

Two things happen to the bags: there are 3 of them, and 5 stones are added. Undo both. Take 5 from each side to get 3x = 15. Then divide by 3 to get x = 5.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The balance is level. Solve 2x + 4 = 14.

    A level balance. The left side has 2 bags of x stones and 4 loose stones, with the words 2x + 4. The right side has 14 loose stones, with the words 14.xx2x + 414
    Tap your answer.

    1. Not this one. You added 4 to 14. But the 4 is added to the 2x, so take 4 away first. Go to M3.8.6. Try another.

    2. Not this one. You divided 14 by 2 and took away 4. The 4 is not divided by itself. Take 4 from 14 first, then divide by 2. Go to M3.8.6. Try another.

    3. Yes, that is right. Take 4 from both sides: 2x = 10. Divide by 2: x = 5. Check: 2 × 5 + 4 = 14.

    See the answer

    5

    Take 4 from both sides: 2x = 10. Divide by 2: x = 5. Check: 2 × 5 + 4 = 14.

    • If you chose A: You added 4 to 14. But the 4 is added to the 2x, so take 4 away first. Go to M3.8.6.
    • If you chose B: You divided 14 by 2 and took away 4. The 4 is not divided by itself. Take 4 from 14 first, then divide by 2. Go to M3.8.6.

  2. Solve 4x − 3 = 17.

    Tap your answer.

    1. Not this one. You took 3 from 17. But 3 was already taken away from 4x, so add 3 to both sides. Go to M3.8.6. Try another.

    2. Yes, that is right. Add 3 to both sides: 4x = 20. Divide by 4: x = 5. Check: 4 × 5 − 3 = 17.

    3. Not this one. You added 3, which is right, but you stopped. This is 4x = 20. Divide both sides by 4. Go to M3.8.6. Try another.

    See the answer

    5

    Add 3 to both sides: 4x = 20. Divide by 4: x = 5. Check: 4 × 5 − 3 = 17.

    • If you chose A: You took 3 from 17. But 3 was already taken away from 4x, so add 3 to both sides. Go to M3.8.6.
    • If you chose C: You added 3, which is right, but you stopped. This is 4x = 20. Divide both sides by 4. Go to M3.8.6.

  3. Solve 2(x + 3) = 14.

    Tap your answer.

    1. Yes, that is right. Divide both sides by 2: x + 3 = 7. Take 3 from both sides: x = 4. Check: 2 × (4 + 3) = 14.

    2. Not this one. This is the value of 2x, not x. You still need to divide both sides by 2. Go to M3.8.6. Try another.

    3. Not this one. You took 3 from 14. But the 2 multiplies the whole bracket. Divide 14 by 2 first, or expand the bracket. Go to M3.8.6. Try another.

    See the answer

    4

    Divide both sides by 2: x + 3 = 7. Take 3 from both sides: x = 4. Check: 2 × (4 + 3) = 14.

    • If you chose B: This is the value of 2x, not x. You still need to divide both sides by 2. Go to M3.8.6.
    • If you chose C: You took 3 from 14. But the 2 multiplies the whole bracket. Divide 14 by 2 first, or expand the bracket. Go to M3.8.6.

  4. The balance is level. The left side is 5 bags of x. The right side is 3 bags of x and 8 loose stones. Solve 5x = 3x + 8.

    A level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8.xxxxx5xxxx3x + 8
    Tap your answer.

    1. Not this one. After you take 3 bags from each side, 2 bags balance 8 stones. That is 2x = 8. Now divide by 2. Go to M3.8.6. Try another.

    2. Not this one. You added 5 and 3, then divided 8 by that. The bags are taken from both sides, 5 − 3 = 2. Go to M3.8.6. Try another.

    3. Yes, that is right. Take 3 bags from both sides: 2x = 8. Divide by 2: x = 4. Check: 5 × 4 = 20 and 3 × 4 + 8 = 20.

    See the answer

    4

    Take 3 bags from both sides: 2x = 8. Divide by 2: x = 4. Check: 5 × 4 = 20 and 3 × 4 + 8 = 20.

    • If you chose A: After you take 3 bags from each side, 2 bags balance 8 stones. That is 2x = 8. Now divide by 2. Go to M3.8.6.
    • If you chose B: You added 5 and 3, then divided 8 by that. The bags are taken from both sides, 5 − 3 = 2. Go to M3.8.6.

  5. What is the right first step to solve 3x + 5 = 20?

    Tap your answer.

    1. Not this one. Then the sides are no longer equal. Whatever you do, do it to every term on both sides. Go to M3.8.6. Try another.

    2. Not this one. The loose number is already added to the bags. To undo adding, take it away. Go to M3.8.6. Try another.

    3. Not this one. Multiplying makes the numbers bigger and does not free x. Undo the adding first. Go to M3.8.6. Try another.

    4. Yes, that is right. We undo the steps backwards. The loose number was added last, so we take it away first. Then we divide.

    See the answer

    Take the loose number away from both sides.

    We undo the steps backwards. The loose number was added last, so we take it away first. Then we divide.

    • If you chose A: Then the sides are no longer equal. Whatever you do, do it to every term on both sides. Go to M3.8.6.
    • If you chose B: The loose number is already added to the bags. To undo adding, take it away. Go to M3.8.6.
    • If you chose C: Multiplying makes the numbers bigger and does not free x. Undo the adding first. Go to M3.8.6.

  6. Leila solves 2x + 1 = 11 and says x = 4. Put 4 back into the equation. What do you find?

    Tap your answer.

    1. Not this one. The right side of the equation is 11. A true answer must make the left side 11, not 9. Go to M3.8.6. Try another.

    2. Yes, that is right. The right answer is x = 5, because 2 × 5 + 1 = 11. Leila forgot to divide 10 by 2.

    3. Not this one. 2x means 2 times x. Put 4 in place of x, then multiply: 2 × 4. Go to M3.8.2. Try another.

    See the answer

    2 × 4 + 1 = 9, not 11. Her answer is wrong.

    The right answer is x = 5, because 2 × 5 + 1 = 11. Leila forgot to divide 10 by 2.

    • If you chose A: The right side of the equation is 11. A true answer must make the left side 11, not 9. Go to M3.8.6.
    • If you chose C: 2x means 2 times x. Put 4 in place of x, then multiply: 2 × 4. Go to M3.8.2.

  7. From an earlier chapter

    Solve x − 6 = 10.

    Tap your answer.

    1. Yes, that is right. Add 6 to both sides. x = 10 + 6 = 16. Check: 16 − 6 = 10.

    2. Not this one. You took 6 from 10. But 6 was already taken away from x, so add 6 to both sides. Go to M3.8.5. Try another.

    3. Not this one. You multiplied 6 by 10. The opposite of taking away is adding. Go to M3.8.5. Try another.

    See the answer

    16

    Add 6 to both sides. x = 10 + 6 = 16. Check: 16 − 6 = 10.

    • If you chose B: You took 6 from 10. But 6 was already taken away from x, so add 6 to both sides. Go to M3.8.5.
    • If you chose C: You multiplied 6 by 10. The opposite of taking away is adding. Go to M3.8.5.

  8. Solve 3x + 5 = 20.

    See the answer

    x = 5

    Take 5 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 × 5 + 5 = 20.

  9. Solve 3(x + 2) = 24.

    See the answer

    x = 6

    Divide both sides by 3: x + 2 = 8. Take 2 from both sides: x = 6. Check: 3 × (6 + 2) = 24.

Keep it

Copy this into your exercise book.

Undo the steps backwards. Take away the loose number first, then divide. Expand a bracket first, or divide by the number outside. Always put the answer back.

Draw: Draw a balance with 3 bags and 5 stones on the left and 20 stones on the right. Draw each step and write x = 5.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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