Representing Other Algebraic Relationships — Practice
Not everything is proportional. Learn to read a pattern, name the rule, and write it as an expression you can trust.
What these questions cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Proportional — Two quantities with a constant ratio: yx = k always.
- Like terms — Terms with exactly the same variable part; only these can be combined.
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- Sequence — An ordered list of numbers. The nth one is an.
Problem 1
Simplify 5x + 3x.
Worked answer
8x
5x + 3x = 8x
Problem 2
Simplify 7a + 4 - 2a.
Worked answer
5a+4
7a - 2a = 5a
Answer: 5a + 4
Problem 3
Expand 3(x + 5).
Worked answer
3x+15
3 · x = 3x and 3 · 5 = 15
3x + 15
Problem 4
Evaluate 4n + 7 when n = 6.
Worked answer
31
4(6) = 24
24 + 7 = 31
Problem 5
What is the next term in 3, 7, 11, 15, …?
Worked answer
19
It grows by 4 each step
15 + 4 = 19
Problem 6
A pattern goes 4, 9, 14, 19, …. Write the rule for the nth term.
Worked answer
5n-1
Step = 5, so the rule contains 5n
At n = 1, 5(1) = 5 but we need 4, so subtract 1
Rule: 5n - 1. Check n = 3: 15 - 1 = 14 ✓
Problem 7
For the pattern 4, 9, 14, 19, …, find the 20th term.
Worked answer
99
5(20) - 1 = 100 - 1 = 99
Problem 8
Expand -2(3x - 7).
Worked answer
-6x+14
-2 · 3x = -6x
-2 · (-7) = +14
-6x + 14
Problem 9
Simplify 4(x + 2) + 3(x - 5).
Worked answer
7x-7
4x + 8 and 3x - 15
4x + 3x = 7x
8 - 15 = -7
7x - 7
Problem 10
Define a ★ b = 3a - b. Compute 5 ★ 2.
Worked answer
13
3(5) - 2 = 15 - 2 = 13
Problem 11
Evaluate 2x2 when x = -3.
Worked answer
18
(-3)2 = 9
2 × 9 = 18
Problem 12
What is the next term in 2, 6, 18, 54, …?
Worked answer
162
Each term is 3 times the one before
54 × 3 = 162
Problem 13
Simplify 5(2x - 3) - 2(x + 4).
Worked answer
8x-23
5(2x - 3) = 10x - 15
-2(x + 4) = -2x - 8 — the minus hits both terms
10x - 2x = 8x and -15 - 8 = -23
8x - 23
Problem 14
Define a ◇ b = a2 - 2b. Compute 4 ◇ (3 ◇ 1).
Worked answer
2
3 ◇ 1 = 32 - 2(1) = 7
4 ◇ 7 = 42 - 2(7) = 16 - 14 = 2
Problem 15
A sequence goes 3, 8, 15, 24, 35, …. Find the 6th term.
Worked answer
48
Differences: 5, 7, 9, 11, so the next difference is 13
35 + 13 = 48
The rule is n(n + 2): at n = 6, 6 × 8 = 48 ✓
Problem 16
Six workers finish a job in 10 days. At the same rate, how many days would 15 workers take?
Worked answer
4
Total work: 6 × 10 = 60 worker-days
60 ÷ 15 = 4 days
Notice the product stays constant: xy = 60.