Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 3, lesson 2, letters and expressions. Work out 6x + 2 when x = 5.
Answer
32. 6 × 5 = 30, then 30 + 2. - From Unit 3, lesson 3, term-to-term and position rules. The position rule is 3n + 1. What is the 7th term?
Answer
22. 3 × 7 = 21, then 21 + 1. - From Unit 1, order of operations. What is 20 − 2 × 6?
Answer
8. Multiply first: 2 × 6 = 12, then 20 − 12 = 8.
Words used in this lesson
- term
- one part of an expression, with its sign. For example: 3x and −y are terms of 3x + 5y + 2x − y.
- like terms
- terms with the same letter part. For example: 3x and 2x.
- unlike terms
- terms with different letter parts. For example: 5x and 4y.
- coefficient
- the number in front of the letter. For example: the 3 in 3x.
- simplify
- write an expression in its shortest equal form. For example: 3x + 2x simplifies to 5x.
- substitute
- put a number in the place of a letter. For example: put x = 2 into 5x.
Terms that match
Each basket of tomatoes holds x tomatoes and each basket of onions holds y onions. A stall has 3 baskets of tomatoes and 5 of onions. A delivery brings 2 more baskets of tomatoes, and one basket of onions is sold. In symbols, 3x + 5y + 2x − y. 3x and 2x both count tomatoes, so they combine to 5x. 5x and 4y cannot become one term, because x and y can change separately. Terms that count the same letter are like terms; the second figure shows why the others cannot join.
Predict first: in 3x + 5y + 2x − y, which terms are like terms? Say what you think the short form is, then step through to see.
3x + 5y + 2x − y has 4 terms. Which of them are like terms?
Show me the steps
- 3x + 5y + 2x − y: the x terms are 3x and 2x, and the y terms are 5y and −y. Sorted: 3x + 2x and 5y − y. Combined: 5x + 4y.
- 4a + 3 + a + 6: the a terms are 4a and a (which is 1a), and the numbers are 3 and 6. Combined: 5a + 9.
- 6m − 2m + 5: 6m − 2m = 4m, and the 5 is on its own. Combined: 4m + 5.
- 2x + 3y − x + y: 2x − x = x (not 0x), and 3y + y = 4y. Combined: x + 4y.
- 7p + 2q − 3p − 2q: 7p − 3p = 4p, and 2q − 2q = 0. The q terms cancel, and the result is 4p.
- 5c + 3d − 2c + 4 + 2d + 1: c terms give 3c, d terms give 5d, the numbers give 5. Combined: 3c + 5d + 5.
Only like terms combine. Like terms have the same letter part: 3x and 2x are like terms, 5y and −y are like terms, and plain numbers are like terms with each other. Add or subtract the numbers in front and keep the letter: 3x + 2x = 5x.
Each term keeps the sign in front of it, so the term after a minus is a subtraction. The letter on its own, such as y, means 1y. Terms with different letters, such as 5x and 4y, stay as they are.
Is the short form really equal?
A simplified expression is only useful if it gives the same value as the original for every value of the letters. Test with numbers: put values of x and y into both and compare.
x = 2, y = 1: the original is 14 and 5x + 4y is 14, the same. 9xy is 18, which is different.
Show me the steps
- 3x + 5y + 2x − y with x = 2 and y = 1: the original is 6 + 5 + 4 − 1 = 14, and 5x + 4y is 10 + 4 = 14. The wrong 9xy is 18.
- x = 4, y = 3: the original is 12 + 15 + 8 − 3 = 32, and 5x + 4y is 20 + 12 = 32. The wrong 9xy is 108.
- x = 0 and y = 0: all three are 0. Matching at one pair proves nothing. The short form must match for every pair, and 9xy does not.
- The second expression, 4x + 3y + x + 6, with x = 2 and y = 3: the original is 8 + 9 + 2 + 6 = 25, and 5x + 3y + 6 is 10 + 9 + 6 = 25. The wrong 14xy is 84.
- The third expression, 7y − 3y + 2x + 5, with x = 3 and y = 2: the original is 14 − 6 + 6 + 5 = 19, and 2x + 4y + 5 is 6 + 8 + 5 = 19. The wrong 11xy is 66.
A common mistake: “3x + 5y = 8xy.” Put x = 2 and y = 1 in: 3x + 5y is 11 but 8xy is 16. x and y can stand for different numbers, so 3x + 5y is already as short as it can be.
Worked example: collecting
Simplify 5x + 3y + 4 + 2x + y.
- The x terms: 5x + 2x = 7x.
- The y terms: 3y + y = 4y. The y on its own is 1y.
- The numbers: just 4.
- Put it together: 7x + 4y + 4.
- Check with x = 2 and y = 1: the original is 10 + 3 + 4 + 4 + 1 = 22, and 7x + 4y + 4 is 14 + 4 + 4 = 22.
Your go. The first step is given. Simplify 9m + 2 − 4m + 5 − m, then check with m = 3.
- The m terms are 9m, −4m and −m. Combine them: 9 − 4 − 1 gives the number in front.
- Now add the plain numbers, write the short form and check it with m = 3 yourself.
Check the answer
9 − 4 − 1 = 4, so 4m. The numbers: 2 + 5 = 7. The short form is 4m + 7. Check: the original is 27 + 2 − 12 + 5 − 3 = 19, and 4 × 3 + 7 = 19.
Check before you go on
1. Which is 4x + 3y + 2x + y written in its shortest form?
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. 5a + 2a is the same as a single term, ka. What is k?
3. Simplify 8x + 3 − 5x + 4, then work out its value when x = 10.
Level 2 · Building
4. Simplify 6p + 4q − 2p + q, then work out its value when p = 3 and q = 2.
5. At a stall, a seller has 3b + 2 oranges in the morning and 5b + 4 in the afternoon, where b is the number of oranges in a full bag. Each bag holds 6 oranges. How many oranges is that in total?
6. Which of these cannot be written any shorter?
Level 3 · Stretch
7. 7x + 3y − 2x − 3y simplifies to a single term kx. What is the value of kx when x = 6?
8. 3x + 4 + 2x + k simplifies to 5x + 9, where k is a plain number. What is k?
Something to think about
Two numbers x and y add up to 10. Without knowing either one, what is the value of 4x + 4y? Test it with two different pairs.
One way to think about it
4x + 4y = 4(x + y) = 4 × 10 = 40. With x = 3 and y = 7: 12 + 28 = 40. With x = 6 and y = 4: 24 + 16 = 40. The answer does not depend on which pair you pick, because both terms are 4 times a letter and the letters add to 10.
Exit ticket
9. Ravi simplifies 3x + 5y as 8xy. Which reply is right?