Simplifying Expressions and Collecting Like Terms

Count how many copies of a are added.

Add the coefficients because the letter parts match.

Subtract the number of copies of b.

Collect the m terms; keep the constant separate.

There is one unwritten coefficient in front of the final q.

The first x has coefficient 1.

1. Combine the coefficients with their signs: 1+4-2=3.
2. Therefore x+4x-2x=3x.

Collect the y terms and the constants separately.

1. 2y-y=y and 6+5=11.
2. Together these give y+11.

The coefficient calculation can have a negative answer.

1. Subtract the coefficients: 4-7=-3.
2. Therefore 4a-7a=-3a.

Match the powers as well as the letter.

1. The squared terms give 3x^2+x^2=4x^2.
2. The other terms give 2x-x=x. The answer is 4x^2+x.

Multiplication can be written in either order: ba=ab.

1. Rewrite 3ba as 3ab.
2. The coefficients give 6+3-2=7, so the answer is 7ab.

Add all three sides, then collect each kind of term.

1. The perimeter is 2x+1+x+4+3x-2.
2. The x terms give 6x and the constants give 1+4-2=3. The perimeter is (6x+3) cm.

Compare the total coefficient of a on each side.

1. The known terms give 5a-3a=2a.
2. The missing term must add another 7a to make 9a, so the missing coefficient is 7.

Look for terms that cancel completely.

1. 5x-5x=0, so no x term remains.
2. 2y+7y=9y. Keep the constant -4: the answer is 9y-4.

The coefficients are decimals, but the collecting rule is the same.

1. The p coefficient is 2.5-0.5=2.
2. The q coefficient is 1.5+2.5=4. The answer is 2p+4q.

Compare adding two copies of a number with multiplying the number by itself.

1. a+a=2a, whereas a\times a=a^2. Eva has confused addition with multiplication.
2. The correct simplification is 2a+3. The constant cannot be collected with the a term.
3. For example, at a=4, the original gives 4+4+3=11, but Eva’s expression gives 4^2+3=19.

Make three groups: squared terms, ordinary x terms and constants.

1. Squared terms: 3x^2-x^2=2x^2.
2. Ordinary x terms: 5x-8x=-3x. Constants: -7+12=5.
3. Together these give 2x^2-3x+5.

Collect only matching terms. To check Leo’s claim, compare both expressions for the given values.

Part (a)

1. Collect a terms: 4a-2a=2a.
2. Collect b terms: 3b+5b=8b. The simplified expression is 2a+8b.

Part (b)

1. At a=2, b=1, the correct expression gives 2\times2+8\times1=12.
2. Leo’s expression gives 10\times2\times1=20.
3. The different values show the expressions are not always equal. Agreement at zero would not prove his claim.

A rectangle has two copies of each side length.

1. Add all four sides: 3p+2+p+5+3p+2+p+5.
2. The letter terms total 8p. The constants total 2+5+2+5=14.
3. The perimeter is (8p+14) cm.

Compare the x coefficients, then compare the y coefficients.

Part (a)

1. The x coefficient on the left is A-2. It must equal 4.
2. Therefore A=4+2=6.

Part (b)

1. The y coefficient on the left is 5-B. It must equal 2.
2. Therefore B=5-2=3. Check: 6x+5y-2x-3y=4x+2y.

Add the three original counts, then subtract both removals.

1. Add the original counts: 4n+3+2n+5+3n-2=9n+6.
2. Subtract the n books given out and the 4 damaged books: 9n+6-n-4.
3. Collect again: (9-1)n+(6-4)=8n+2.