Chapter 3 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

M3.8.3 Chapter 3

Collecting like terms

Sipho has 3 boxes of mangoes and 2 boxes of limes. A mango box holds a mangoes. A lime box holds b limes. He gets 5 more mango boxes and gives away 1 lime box. How many pieces of fruit does he have?

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

  • Say which terms are like terms and which are not
  • Simplify 3a + 2b + 5a − b

Sipho has 3 boxes of mangoes and 2 boxes of limes. Each mango box holds a mangoes. Each lime box holds b limes. He has 3a + 2b pieces of fruit.

Each part of an expression, with the sign in front of it, is a term. The expression 3a + 2b has two terms. They are 3a and 2b.

Three rows of bars. In the first row, 2 bars of a and 3 bars of a, with the words 2a and 3a are like terms. In the second row, 2 bars of a and 3 hatched bars of b, with the words 2a and 3b: not like terms. In the third row, 2 bars of a and 3 small squares, with the words 2a and 3: not like terms.aaaaa2a and 3a are like termsaabbb2a and 3b: not like termsaa2a and 3: not like terms
The shading tells the letters apart. The length of a bar is only for the picture.

The numbers a and b may be different, so 3a and 2b cannot be joined. But 3a and 5a are both a number of a. Terms with the same letter part, such as 3a and 5a, are like terms. The letter part of a² is not the same as a.

We can count like terms together. We cannot count terms with different letters together. A plain number is not like a term with a letter.

Draw it first

Group the bars that are alike

  1. Step 1: four rows of bars: 3 bars of a, 2 bars of b, 5 bars of a, and 1 crossed out bar of b, with the words 3a + 2b + 5a − b. Step 2: the a bars together and the b bars together, with the words like terms together. Step 3: 8 bars of a and 1 bar of b, with the words 8a + b.aaabbaaaaab3a + 2b + 5a − b

    1 Draw one row for each term: 3a + 2b + 5a − b. The crossed bar is taken away.

  2. Step 1: four rows of bars: 3 bars of a, 2 bars of b, 5 bars of a, and 1 crossed out bar of b, with the words 3a + 2b + 5a − b. Step 2: the a bars together and the b bars together, with the words like terms together. Step 3: 8 bars of a and 1 bar of b, with the words 8a + b.aaaaaaaabbblike terms together

    2 Move the bars that are alike next to each other.

  3. Step 1: four rows of bars: 3 bars of a, 2 bars of b, 5 bars of a, and 1 crossed out bar of b, with the words 3a + 2b + 5a − b. Step 2: the a bars together and the b bars together, with the words like terms together. Step 3: 8 bars of a and 1 bar of b, with the words 8a + b.aaaaaaaab8a + b

    3 Count each kind. There are 8 bars of a. The crossed bar cancels 1 of the 2 bars of b, so 1 bar of b is left.

We add the numbers in front of the letters and keep the letter. 3a + 5a = 8a. And 2b − b = b. We write b, not 1b.

Worked example

Simplify 3a + 2b + 5a − b.

  1. The a terms are 3a and 5a. The b terms are 2b and −b.
  2. Count the a bars: 3 + 5 = 8.
  3. Count the b bars: 2 − 1 = 1.
  4. So 3a + 2b + 5a − b = 8a + b.

Answer: 8a + b

To simplify an expression is to write it in its shortest form. We collect the like terms to do this.

Worked example

Simplify 4x + 3 + 2x − 1.

  1. The x terms are 4x and 2x. The plain numbers are 3 and −1.
  2. Count the x bars: 4 + 2 = 6.
  3. Add the plain numbers: 3 − 1 = 2.
  4. So 4x + 3 + 2x − 1 = 6x + 2.

Answer: 6x + 2

The term a² is not like the term a. The term a² is a square. The term a is a bar. The two shapes are different.

On the left, a hatched square with side a on the top and on the left, and a² inside. On the right, one long bar labelled a.aaa²a
a² and a are not like terms.

Worked example

Check that 8a + b is the same as 3a + 2b + 5a − b when a = 2 and b = 3.

  1. Put a = 2 and b = 3 into 3a + 2b + 5a − b.
  2. The a terms: 3 × 2 + 5 × 2 = 16.
  3. The b terms: 2 × 3 − 3 = 3.
  4. Add them: 16 + 3 = 19.
  5. Now 8a + b gives 8 × 2 + 3 = 19. The two agree.

Answer: 19 and 19. They agree.

See it three ways

With things

Put 2 cups of beans and 3 more cups of beans in a row. Every cup holds the same number of beans, a. Count the cups. 5 cups of a beans hold 5a beans.

As a picture

Five bars of a in a row, with the words 2a + 3a = 5a under them.aaaaa2a + 3a = 5a

In symbols

2a + 3a = 5a. Add the numbers in front. The letter stays.

Try it

Cups and bowls

You need: 3 cups and 2 bowls, beans or stones to put in them, paper and a pencil.

  1. Put the same number of beans in each cup. Put the same number of stones in each bowl. Do not count them.
  2. Let a be the beans in a cup and b the stones in a bowl. Put 3 cups and 2 bowls on the table.
  3. Add 4 more cups. Take away 1 bowl.
  4. Count the cups and the bowls. Write the number of beans and stones with a and b.

What you will see: There are 7 cups of a beans and 1 bowl of b stones. This is 3a + 2b + 4a − b = 7a + b in all. The numbers a and b may differ, so they stay apart.

If you cannot try it, this is what happens.
Seven bars of a in the first row and one hatched bar of b in the second row, with the words 7a + b under them.aaaaaaab7a + b

A wrong idea many people have

Always add the numbers in front. So 3a + 2b = 5ab.

Why it feels right: 3 + 2 is 5, and it seems tidy to keep both letters.

The test that breaks it: Let a be 10 and b be 1. Then 3a + 2b is 30 + 2 = 32. But 5ab is 5 × 10 × 1 = 50. They are not equal.

What is true: Only like terms are joined. 3a + 2b has two different kinds of term, so it stays 3a + 2b.

Key words

term
one part of an expression, with the sign in front of it.
like terms
terms with the same letter part, such as 3a and 5a.
simplify
to write an expression in its shortest form.

Back to your guess

The answer is B: 8a + b. The mango boxes are counted together, and the lime boxes are counted together.

The numbers a and b may be different, so 3a and 2b cannot be joined. He has 3 + 5 = 8 boxes of a mangoes and 2 − 1 = 1 box of b limes. That is 8a + b pieces of fruit.

Check yourself

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

  1. Which pair is a pair of like terms?

    Tap your answer.

    1. Not this one. The letters are different. 4x is a number of x, and 7y is a number of y. Go to M3.8.3. Try another.

    2. Yes, that is right. Both terms have the letter x and nothing else. The numbers in front may differ.

    3. Not this one. The 7 is a plain number. It has no x, so it is not like 4x. Go to M3.8.3. Try another.

    4. Not this one. x² is a square, but x is a bar. They are different shapes, so they are not like terms. Go to M3.8.3. Try another.

    See the answer

    4x and 7x

    Both terms have the letter x and nothing else. The numbers in front may differ.

    • If you chose A: The letters are different. 4x is a number of x, and 7y is a number of y. Go to M3.8.3.
    • If you chose C: The 7 is a plain number. It has no x, so it is not like 4x. Go to M3.8.3.
    • If you chose D: x² is a square, but x is a bar. They are different shapes, so they are not like terms. Go to M3.8.3.

  2. Simplify 5a + 3a.

    Tap your answer.

    1. Yes, that is right. 5 bars of a and 3 bars of a make 8 bars of a. So 5a + 3a = 8a.

    2. Not this one. You multiplied 5 × 3. When you collect like terms, you add the numbers in front. Go to M3.8.3. Try another.

    3. Not this one. The letter stays. 5 bars of a and 3 more bars of a are 8 bars of a, not just 8. Go to M3.8.3. Try another.

    4. Not this one. a + a is 2a, not a square. When we add like terms, the letter part does not change. Go to M3.8.1. Try another.

    See the answer

    8a

    5 bars of a and 3 bars of a make 8 bars of a. So 5a + 3a = 8a.

    • If you chose B: You multiplied 5 × 3. When you collect like terms, you add the numbers in front. Go to M3.8.3.
    • If you chose C: The letter stays. 5 bars of a and 3 more bars of a are 8 bars of a, not just 8. Go to M3.8.3.
    • If you chose D: a + a is 2a, not a square. When we add like terms, the letter part does not change. Go to M3.8.1.

  3. Simplify 4x + 3y + 2x − 2y.

    Tap your answer.

    1. Not this one. You added 3y and 2y. But the sign in front of 2y is minus, so it is taken away. Go to M3.8.3. Try another.

    2. Yes, that is right. The x terms: 4x + 2x = 6x. The y terms: 3y − 2y = y. So the answer is 6x + y.

    3. Not this one. You joined everything into one term. The x terms and the y terms stay apart. Go to M3.8.3. Try another.

    4. Not this one. You took the 3y away from the 2y. But we start with 3y, then take 2y away. That leaves 1y. Go to M3.8.3. Try another.

    See the answer

    6x + y

    The x terms: 4x + 2x = 6x. The y terms: 3y − 2y = y. So the answer is 6x + y.

    • If you chose A: You added 3y and 2y. But the sign in front of 2y is minus, so it is taken away. Go to M3.8.3.
    • If you chose C: You joined everything into one term. The x terms and the y terms stay apart. Go to M3.8.3.
    • If you chose D: You took the 3y away from the 2y. But we start with 3y, then take 2y away. That leaves 1y. Go to M3.8.3.

  4. Simplify 7x + 4 − 3x + 1.

    Tap your answer.

    1. Not this one. The last term is plus 1, not minus 1. The plain numbers are 4 and 1, which make 5. Go to M3.8.3. Try another.

    2. Not this one. The sign in front of 3x is minus. So 7x − 3x is 4x. Go to M3.8.3. Try another.

    3. Not this one. You joined the plain numbers to the x terms. A plain number is not like a term with x. Go to M3.8.3. Try another.

    4. Yes, that is right. The x terms: 7x − 3x = 4x. The plain numbers: 4 + 1 = 5. So the answer is 4x + 5.

    See the answer

    4x + 5

    The x terms: 7x − 3x = 4x. The plain numbers: 4 + 1 = 5. So the answer is 4x + 5.

    • If you chose A: The last term is plus 1, not minus 1. The plain numbers are 4 and 1, which make 5. Go to M3.8.3.
    • If you chose B: The sign in front of 3x is minus. So 7x − 3x is 4x. Go to M3.8.3.
    • If you chose C: You joined the plain numbers to the x terms. A plain number is not like a term with x. Go to M3.8.3.

  5. The first row is 2a + 3b. The second row is 3a − 2b, and the crossed bars are taken away. What is the whole picture, simplified?

    The first row has 2 bars of a and 3 hatched bars of b. The second row has 3 bars of a and 2 hatched bars of b that are dashed and crossed out.aabbbaaabb
    Tap your answer.

    1. Not this one. You counted the crossed bars of b. They are taken away, so they cancel 2 of the 3 bars. Go to M3.8.3. Try another.

    2. Not this one. You joined the letters. Bars of a and bars of b are different, so they stay apart. Go to M3.8.3. Try another.

    3. Yes, that is right. The a bars: 2 + 3 = 5. The b bars: 3 − 2 = 1. So the answer is 5a + b.

    See the answer

    5a + b

    The a bars: 2 + 3 = 5. The b bars: 3 − 2 = 1. So the answer is 5a + b.

    • If you chose A: You counted the crossed bars of b. They are taken away, so they cancel 2 of the 3 bars. Go to M3.8.3.
    • If you chose B: You joined the letters. Bars of a and bars of b are different, so they stay apart. Go to M3.8.3.

  6. Look at the picture. Which sentence is true about a and a²?

    On the left, a hatched square with side a on the top and on the left, and a² inside. On the right, one long bar labelled a.aaa²a
    Tap your answer.

    1. Not this one. a² is a times a, a square. The term a is a bar. The shapes differ, so they are not like terms. Go to M3.8.3. Try another.

    2. Yes, that is right. Like terms have the same letter part. The terms a and a² have different letter parts. They cannot join into one term.

    3. Not this one. They are different shapes. If a is 3, then a is 3 but a² is 9. Go to M3.8.1. Try another.

    See the answer

    They are not like terms. One is a bar and the other is a square.

    Like terms have the same letter part. The terms a and a² have different letter parts. They cannot join into one term.

    • If you chose A: a² is a times a, a square. The term a is a bar. The shapes differ, so they are not like terms. Go to M3.8.3.
    • If you chose C: They are different shapes. If a is 3, then a is 3 but a² is 9. Go to M3.8.1.

  7. Collect the like terms in 3a + 2b + 5a. What is the number in front of a?

    See the answer

    8

    The a terms are 3a and 5a. 3 + 5 = 8, so the answer is 8a + 2b.

  8. In 2b + 6b − 3b, how many bars of b are left after collecting?

    See the answer

    5

    2 + 6 = 8, and 8 − 3 = 5. So 2b + 6b − 3b = 5b.

  9. From an earlier chapter

    Find the value of 3a + 2b when a = 4 and b = 5.

    Tap your answer.

    1. Not this one. You added every number: 3 + 4 + 2 + 5. A number next to a letter means multiply. Go to M3.8.2. Try another.

    2. Not this one. You put the numbers side by side: 34 and 25. 3a means 3 × 4, and 2b means 2 × 5. Go to M3.8.2. Try another.

    3. Yes, that is right. 3a is 3 × 4 = 12 and 2b is 2 × 5 = 10. Then 12 + 10 = 22.

    See the answer

    22

    3a is 3 × 4 = 12 and 2b is 2 × 5 = 10. Then 12 + 10 = 22.

    • If you chose A: You added every number: 3 + 4 + 2 + 5. A number next to a letter means multiply. Go to M3.8.2.
    • If you chose B: You put the numbers side by side: 34 and 25. 3a means 3 × 4, and 2b means 2 × 5. Go to M3.8.2.

Keep it

Copy this into your exercise book.

Like terms have the same letter part. Collect them: 3a + 5a = 8a. A letter alone, such as b, means 1b. Different letters stay apart.

Draw: Draw bars for 3a + 2b + 5a − b. Group the a bars and the b bars. Write 8a + b.

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