HSPT: Expressions: distribute and combine
Learn to evaluate and simplify expressions with parentheses, by substituting or by distributing and combining like terms.
How do you work out an expression like 2(x + 7) − 3x without slipping up?
Short answer
Either substitute the value and do the parentheses first, or distribute to every term inside and combine like terms. Both routes should give the same answer.
Know first: Order of operations, Adding and subtracting negative numbers
Step 1 of 5
Two routes to the same value
Expressions like 2(x + 7) − 3x can be handled two ways. Substitute first: put in the value of x and follow the order of operations, parentheses first. Simplify first: distribute, combine like terms, then substitute.
Both routes give the same number. Simplifying first is worth it when the question asks for an equal expression, or when x is awkward to plug in.
| Route | Work | Value |
|---|---|---|
| Substitute first | 2(5 + 7) − 3(5) = 24 − 15 | 9 |
| Simplify first | 2x + 14 − 3x = −x + 14, then −5 + 14 | 9 |
If your two routes disagree, one of them has a slip.
Step 2 of 5
Substitute, then follow the order of operations
Worked example: Evaluate an expression
If x = 5, what is 2(x + 7) − 3x?
- A9
- B2
- C−6
- D24
- 1
Parentheses first
x + 7 = 5 + 7 = 12.
- 2
Multiply
2 × 12 = 24, and 3x = 3 × 5 = 15.
- 3
Subtract
24 − 15 = 9.
- 4
See why the other choices are there
2 multiplies only the x: 2(5) + 7 − 15. −6 subtracts inside the parentheses before multiplying. 24 forgets to subtract 3x.
Answer
9
Find the wrong step
A student evaluates 3(x − 1) − 2x for x = 9 by simplifying first.
- 1
Distributes: 3x − 1 − 2x.
What went wrong
The 3 multiplies both terms inside, so 3(x − 1) = 3x − 3.
- 2
Combines: x − 1.
- 3
Substitutes: 9 − 1 = 8.
The fix
3x − 3 − 2x = x − 3, and 9 − 3 = 6. Check by substituting first: 3(8) − 18 = 6.
Step 3 of 5
Choose a route and check with the other
Check yourself · Question 1
If x = 6, what is 5(x + 1) − 2x?
Answer: B
Parentheses: 6 + 1 = 7. Then 5 × 7 = 35 and 2x = 12, so 35 − 12 = 23. Simplifying first agrees: 5x + 5 − 2x = 3x + 5 = 23.
Trap answer: 19
Distributing to one term
- Why it looks right
- The 5 sits right next to the x, so multiplying only x feels complete.
- Why it's wrong
- The 5 multiplies the whole group, x + 1. So 5(x + 1) is 5x + 5, which is 35 when x = 6.
- The right answer
- 23
Check yourself · Question 2
Which expression equals 5(x + 3) − 7x for every value of x?
Answer: A
Distribute: 5x + 15 − 7x. Combine like terms: 5x − 7x = −2x. The expression is −2x + 15. Test with x = 1: 5(1 + 3) − 7(1) = 20 − 7 = 13, and −2(1) + 15 = 13.
Common mistake
"5(x + 1) means 5x + 1."
- Why it's tempting
- The 5 is written right next to x, so it looks like it belongs only to x.
- Do this instead
- Draw an arrow from the outside number to each term in the parentheses. Every term gets multiplied.
Step 4 of 5
What to remember
Remember
- Substituting first means parentheses first, then multiply, then add and subtract.
- The number outside parentheses multiplies every term inside.
- Combine like terms carefully with their signs, and check one route with the other.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
Which route did you use in the practice question, and what would you get if the outside number multiplied only x?