Chapter 5 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

M3.8.5 Chapter 5

Solving equations in one step

A balance is level. The left pan holds a bag of x stones and 7 loose stones. The right pan holds 12 loose stones. How many stones are in the 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

  • Use a balance picture to keep both sides equal
  • Solve x + 7 = 12 and 4x = 20

A balance is level when both sides weigh the same. Every stone weighs the same. A bag holds x stones, and we do not know x yet. The bag itself weighs nothing.

The left pan holds the bag and 7 loose stones, so it weighs x + 7. The right pan holds 12 loose stones. We write x + 7 = 12.

This is an equation. It says that two things are equal. To solve it is to find the number that x stands for.

Draw it first

Take the same from both sides

  1. Step 1: a level balance. The left side has a bag of x stones and 7 loose stones, with the words x + 7. The right side has 12 loose stones, with the words 12. Step 2: the same balance with 7 stones taken from each side. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Step 3: the words x = 5.xx + 712

    1 The balance is level. The left side is x + 7 and the right side is 12.

  2. Step 1: a level balance. The left side has a bag of x stones and 7 loose stones, with the words x + 7. The right side has 12 loose stones, with the words 12. Step 2: the same balance with 7 stones taken from each side. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Step 3: the words x = 5.xx5

    2 Take 7 stones from each side. The balance stays level, and the bag is on its own.

  3. Step 1: a level balance. The left side has a bag of x stones and 7 loose stones, with the words x + 7. The right side has 12 loose stones, with the words 12. Step 2: the same balance with 7 stones taken from each side. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Step 3: the words x = 5.xx5x = 5

    3 The bag balances 5 stones, so x = 5.

Here is the rule. Whatever you do to one side, do to the other side. Then the two sides stay equal.

Worked example

Solve x + 7 = 12.

  1. The 7 is added to x. The opposite of adding 7 is taking away 7.
  2. Take 7 from both sides: x = 12 − 7 = 5.
  3. So x = 5.
  4. Check: put 5 back into x + 7. We get 5 + 7 = 12. It is true.

Answer: x = 5

Adding and taking away undo each other. Multiplying and dividing undo each other. Each one is the inverse operation of the other.

Draw it first

Share both sides into 4 equal groups

  1. Step 1: a level balance. The left side has 4 bags of x stones, with the words 4x. The right side has 20 loose stones, with the words 20. Step 2: each side shared into 4 equal groups. One group on the left is one bag, with the words x. One group on the right is 5 stones, with the words 5. Under the balance, the 20 stones are drawn as 4 groups of 5, with the words 4 groups of 5. Step 3: the words x = 5.xxxx4x20

    1 Four bags balance 20 stones. This is 4x = 20.

  2. Step 1: a level balance. The left side has 4 bags of x stones, with the words 4x. The right side has 20 loose stones, with the words 20. Step 2: each side shared into 4 equal groups. One group on the left is one bag, with the words x. One group on the right is 5 stones, with the words 5. Under the balance, the 20 stones are drawn as 4 groups of 5, with the words 4 groups of 5. Step 3: the words x = 5.xx54 groups of 5

    2 Share each side into 4 equal groups. The 20 stones make 4 groups of 5. One bag balances 5 stones.

  3. Step 1: a level balance. The left side has 4 bags of x stones, with the words 4x. The right side has 20 loose stones, with the words 20. Step 2: each side shared into 4 equal groups. One group on the left is one bag, with the words x. One group on the right is 5 stones, with the words 5. Under the balance, the 20 stones are drawn as 4 groups of 5, with the words 4 groups of 5. Step 3: the words x = 5.xx54 groups of 5x = 5

    3 So x = 5.

Worked example

Solve 4x = 20.

  1. 4x means 4 × x. The opposite of multiplying by 4 is dividing by 4.
  2. Divide both sides by 4: 4x ÷ 4 = 20 ÷ 4.
  3. Work out 20 ÷ 4 = 5. So x = 5.
  4. Check: put 5 back into 4x. We get 4 × 5 = 20. It is true.

Answer: x = 5

What if something is taken away from x? Look at x − 3 = 8. The whole x is cut into two parts. One part is 3, and the other part is 8.

One long bar with the word x over it. It is cut into two parts. The left part is 8 and the right part is 3.83x
x − 3 = 8. Put the 3 back, and the whole bar is x.

Worked example

Solve x − 3 = 8.

  1. The 3 is taken away from x. The opposite of taking away 3 is adding 3.
  2. Add 3 to both sides of x − 3 = 8.
  3. On the right side: 8 + 3 = 11.
  4. So x = 11.
  5. Check: put 11 back into x − 3. We get 11 − 3 = 8. It is true.

Answer: x = 11

See it three ways

With things

Count out 12 stones. Hide 7 of them in your hand. Count the stones left on the table.

As a picture

One long bar with the number 12 over it. It is cut into two parts. The left part is x, shaded, and the right part is 7.x712

In symbols

x + 7 = 12, so x = 12 − 7 = 5.

Try it

Make your own balance

You need: a ruler or a flat stick, a pencil or a round bottle to rest it on, 2 cups that are the same, about 50 small stones or washers, all the same size.

  1. Rest the stick across the pencil so that it can tip. Ask a friend to hide some stones in one cup.
  2. Put that cup and 7 loose stones on the left. Put the empty cup on the right, so the two cups cancel. Add loose stones to the right until the stick is level.
  3. Take 7 stones from the left. Take 7 stones from the right. Is the stick still level?
  4. Now the stones in the cup balance the stones on the right. Count them. That is x.

What you will see: The two cups cancel. The stick stays level because you take the same from both sides. The stones left on the right tell you how many were in the cup.

If you cannot try it, this is what happens.
A level balance. The left side is one bag of x stones, with the words x. The right side is 5 loose stones, with the words 5.xx5

A wrong idea many people have

x + 7 = 12, so x = 12 + 7 = 19.

Why it feels right: There is a 7 and a 12 and a plus sign, and you are used to adding.

The test that breaks it: Put 19 back into x + 7. You get 19 + 7 = 26, not 12. So 19 is not the answer.

What is true: Use the opposite operation. The 7 is added to x, so take 7 from both sides.

Key words

equation
two things that are equal, with a letter for an unknown number.
solve
to find the number that makes an equation true.
inverse operation
the operation that undoes another, such as taking away and adding.

Back to your guess

The answer is A: x = 5. Take 7 stones from both sides, and the bag balances 5 stones.

The pans stay level if we take the same from both sides. Take 7 stones from each pan. The bag is left on its own, and it balances 12 − 7 = 5 stones.

Check yourself

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

  1. The balance is level. Solve x + 3 = 9.

    A level balance. The left side has a bag of x stones and 3 loose stones, with the words x + 3. The right side has 9 loose stones, with the words 9.xx + 39
    Tap your answer.

    1. Not this one. You added 3 and 9. The 3 is added to x, so take 3 from both sides. Go to M3.8.5. Try another.

    2. Yes, that is right. Take 3 from both sides. The bag balances 9 − 3 = 6 stones. Check: 6 + 3 = 9.

    3. Not this one. That is the 3 loose stones on the left. The number x is what the bag holds. Go to M3.8.5. Try another.

    See the answer

    6

    Take 3 from both sides. The bag balances 9 − 3 = 6 stones. Check: 6 + 3 = 9.

    • If you chose A: You added 3 and 9. The 3 is added to x, so take 3 from both sides. Go to M3.8.5.
    • If you chose C: That is the 3 loose stones on the left. The number x is what the bag holds. Go to M3.8.5.

  2. Solve 3x = 18.

    Tap your answer.

    1. Not this one. You took 3 away from 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5. Try another.

    2. Not this one. You added 3 to 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5. Try another.

    3. Not this one. You multiplied 3 by 18. The 3 is already multiplying x, so divide both sides by 3. Go to M3.8.5. Try another.

    4. Yes, that is right. Divide both sides by 3. x = 18 ÷ 3 = 6. Check: 3 × 6 = 18.

    See the answer

    6

    Divide both sides by 3. x = 18 ÷ 3 = 6. Check: 3 × 6 = 18.

    • If you chose A: You took 3 away from 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5.
    • If you chose B: You added 3 to 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5.
    • If you chose C: You multiplied 3 by 18. The 3 is already multiplying x, so divide both sides by 3. Go to M3.8.5.

  3. Solve x − 5 = 9.

    Tap your answer.

    1. Not this one. You took 5 from 9. But 5 was already taken away from x. The opposite is to add 5. Go to M3.8.5. Try another.

    2. Not this one. That is the answer to x − 5, not the number x. Add 5 to both sides to find x. Go to M3.8.5. Try another.

    3. Yes, that is right. Add 5 to both sides. x = 9 + 5 = 14. Check: 14 − 5 = 9.

    See the answer

    14

    Add 5 to both sides. x = 9 + 5 = 14. Check: 14 − 5 = 9.

    • If you chose A: You took 5 from 9. But 5 was already taken away from x. The opposite is to add 5. Go to M3.8.5.
    • If you chose B: That is the answer to x − 5, not the number x. Add 5 to both sides to find x. Go to M3.8.5.

  4. Solve x ÷ 4 = 3.

    Tap your answer.

    1. Not this one. You added 4 and 3. The x is divided by 4, so the opposite is to multiply by 4. Go to M3.8.5. Try another.

    2. Yes, that is right. Multiply both sides by 4. x = 3 × 4 = 12. Check: 12 ÷ 4 = 3.

    3. Not this one. You divided 3 by 4. But x was divided by 4, so multiply both sides by 4. Go to M3.8.5. Try another.

    See the answer

    12

    Multiply both sides by 4. x = 3 × 4 = 12. Check: 12 ÷ 4 = 3.

    • If you chose A: You added 4 and 3. The x is divided by 4, so the opposite is to multiply by 4. Go to M3.8.5.
    • If you chose C: You divided 3 by 4. But x was divided by 4, so multiply both sides by 4. Go to M3.8.5.

  5. The balance is level. Which move keeps it level?

    A level balance. The left side has a bag of x stones and 4 loose stones, with the words x + 4. The right side has 10 loose stones, with the words 10.xx + 410
    Tap your answer.

    1. Not this one. Then the left side is lighter and the balance tips. Take the same from the other side too. Go to M3.8.5. Try another.

    2. Not this one. Then the left side is lighter and the balance tips. Whatever you do on one side, do on the other. Go to M3.8.5. Try another.

    3. Not this one. This makes the sides further apart. Do the same to both sides. Go to M3.8.5. Try another.

    4. Yes, that is right. Equal things stay equal when you take the same from each. This is the one rule of solving equations.

    See the answer

    Take the same number of stones from both sides.

    Equal things stay equal when you take the same from each. This is the one rule of solving equations.

    • If you chose A: Then the left side is lighter and the balance tips. Take the same from the other side too. Go to M3.8.5.
    • If you chose B: Then the left side is lighter and the balance tips. Whatever you do on one side, do on the other. Go to M3.8.5.
    • If you chose C: This makes the sides further apart. Do the same to both sides. Go to M3.8.5.

  6. Tariq solves x + 8 = 15 and says x = 23. Test his answer by putting it back into x + 8. What do you find?

    Tap your answer.

    1. Yes, that is right. The correct answer takes 8 from 15: x = 15 − 8 = 7. Check: 7 + 8 = 15. Tariq added the 8 instead.

    2. Not this one. The equation says x + 8, not x − 8. Put 23 into x + 8. Go to M3.8.5. Try another.

    3. Not this one. The right side of the equation is 15. A true answer must give 15, not 31. Go to M3.8.5. Try another.

    See the answer

    23 + 8 = 31, not 15. His answer is wrong.

    The correct answer takes 8 from 15: x = 15 − 8 = 7. Check: 7 + 8 = 15. Tariq added the 8 instead.

    • If you chose B: The equation says x + 8, not x − 8. Put 23 into x + 8. Go to M3.8.5.
    • If you chose C: The right side of the equation is 15. A true answer must give 15, not 31. Go to M3.8.5.

  7. From an earlier chapter

    Expand 2(x + 3).

    Tap your answer.

    1. Not this one. You multiplied only the first term. The 2 multiplies the 3 as well. Go to M3.8.4. Try another.

    2. Yes, that is right. 2 × x = 2x and 2 × 3 = 6. So 2(x + 3) = 2x + 6.

    3. Not this one. You multiplied only the second term. The 2 multiplies the x as well. Go to M3.8.4. Try another.

    See the answer

    2x + 6

    2 × x = 2x and 2 × 3 = 6. So 2(x + 3) = 2x + 6.

    • If you chose A: You multiplied only the first term. The 2 multiplies the 3 as well. Go to M3.8.4.
    • If you chose C: You multiplied only the second term. The 2 multiplies the x as well. Go to M3.8.4.

  8. Solve x + 9 = 20.

    See the answer

    x = 11

    Take 9 from both sides. x = 20 − 9 = 11. Check: 11 + 9 = 20.

  9. Solve 6x = 42.

    See the answer

    x = 7

    Divide both sides by 6. x = 42 ÷ 6 = 7. Check: 6 × 7 = 42.

Keep it

Copy this into your exercise book.

An equation is a level balance. Do the same to both sides. Use the inverse operation: take away to undo add, divide to undo multiply. Always put the answer back.

Draw: Draw a balance. Put x and 7 stones on the left and 12 stones on the right. Cross out 7 from each side. 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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