Representing Other Algebraic Relationships — Challenge
Not everything is proportional. Learn to read a pattern, name the rule, and write it as an expression you can trust.
What these problems 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
What is the value of 3a+2 when a=5?
- 10
- 15
- 17
- 25
- 32
Worked answer
17
Substitute a=5, remembering that multiplication comes before addition.
3· 5+2 = 15+2 = 17. Adding first would wrongly give 3· 7 = 21.
Problem 2
Simplify 4x+3-2x+5.
- 2x+8
- 6x+8
- 2x+2
- 6x+2
- 2x-8
Worked answer
2x+8
Collect the x terms: 4x-2x = 2x.
Collect the constants: 3+5 = 8.
The expression simplifies to 2x+8.
Problem 3
If 2x+3y=12 and x=3, what is y?
- 1
- 2
- 3
- 4
- 6
Worked answer
2
Substitute x=3: 2(3)+3y = 12, so 6+3y = 12.
Subtract 6: 3y = 6, so y = 2.
Problem 4
What is the value of x2-4 when x=3?
- -1
- 2
- 5
- 9
- 13
Worked answer
5
Square first, then subtract: 32 = 9.
9-4 = 5. Subtracting before squaring would give (3-4)2 = 1, a different expression.
Problem 5
Expand 3(2x-5).
- 6x-15
- 6x-5
- 5x-15
- 6x+15
- 2x-15
Worked answer
6x-15
Multiply both terms inside the bracket by 3.
3· 2x = 6x and 3·(-5) = -15, giving 6x-15.
Multiplying only the first term is the common slip and gives 6x-5.
Problem 6
A rectangle has length x+4 and width x, and its perimeter is 28. What is x?
- 3
- 4
- 5
- 6
- 10
Worked answer
5
The perimeter is twice the length plus twice the width: 2(x+4)+2x.
That simplifies to 4x+8, and setting it equal to 28 gives 4x = 20, so x = 5.
Check: the sides are 9 and 5, and 2(9)+2(5) = 28.
Problem 7
What is the value of 2(a+b) when a=3 and b=4?
- 7
- 10
- 11
- 14
- 24
Worked answer
14
Work out the bracket first: a+b = 3+4 = 7.
Then multiply: 2· 7 = 14.
Multiplying only a by 2 would give 2(3)+4 = 10, which is a different expression.
Problem 8
Factorise 6x+9 completely.
- 3(2x+3)
- 3(2x+9)
- 6(x+9)
- 9(x+1)
- 3(x+3)
Worked answer
3(2x+3)
Find the greatest common factor of 6 and 9, which is 3.
Divide each term by 3: 6x÷ 3 = 2x and 9÷ 3 = 3.
So 6x+9 = 3(2x+3). Expanding it back gives 6x+9, which checks the answer.
Problem 9
If y=2x-1, what is x when y=9?
- 4
- 4.5
- 5
- 17
- 19
Worked answer
5
Substitute y=9: 9 = 2x-1.
Add 1 to both sides: 2x = 10, so x = 5.
Check: 2(5)-1 = 9.
Problem 10
Which expression means five more than twice a number n?
- 2n+5
- 5n+2
- 2(n+5)
- 5-2n
- n+10
Worked answer
2n+5
Twice the number is 2n.
Five more than that adds 5 afterwards, giving 2n+5.
The option 2(n+5) means twice the result of adding five first, which is 2n+10 and so is different.