Year 7 Algebraic techniques: free practice test with answers

Sample algebraic techniques questions with answers

Try each one before you open the answer.

1. Introduction to algebra

   Write an algebraic expression for each of the following.

   (a) $5$ more than $x$

   (b) the product of $4$ and $m$

   (c) half of $k$

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   Key idea: Translate each phrase into operations: more than means add, product means multiply, half means divide by $2$.

   1. (a) $5$ more than $x$: $x + 5$
   2. (b) $4 \times m$ is written $4m$
   3. (c) $k \div 2$ is written $\dfrac{k}{2}$

   Answer: (a) $x + 5$ (b) $4m$ (c) $\dfrac{k}{2}$

2. Substituting numbers into algebraic expressions

   If $a = 6$, evaluate $3a + 7$.

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   Key idea: Replace the pronumeral with its value, then follow the order of operations.

   1. $3a + 7 = 3 \times 6 + 7$
   2. $= 18 + 7 = 25$

   Answer: $25$

3. Equivalent algebraic expressions

   Which of these expressions are equivalent to $3x + 6$?

   A. $3(x + 2)$ B. $6 + 3x$ C. $3x + 2$ D. $x + x + x + 6$

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   Key idea: Equivalent expressions give the same value for every value of the pronumeral.

   1. A: $3(x + 2) = 3 \times x + 3 \times 2 = 3x + 6$ ✓
   2. B: $6 + 3x$ is $3x + 6$ in a different order ✓
   3. C: $3x + 2$ has a different constant. For $x = 1$: $5$ compared with $3x + 6 = 9$ ✗
   4. D: $x + x + x = 3x$, so $3x + 6$ ✓

   Answer: A, B and D

4. Like terms

   Simplify $5a + 3b - 2a + b$.

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   Key idea: Like terms have exactly the same pronumerals; only like terms can be added or subtracted.

   1. Group like terms: $5a - 2a + 3b + b$
   2. $5a - 2a = 3a$ and $3b + b = 4b$
   3. Result: $3a + 4b$

   Answer: $3a + 4b$

5. Multiplying and dividing expressions

   Simplify $4a \times 3b$.

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   Key idea: Multiply the numbers together and write the pronumerals next to each other in alphabetical order.

   1. Numbers: $4 \times 3 = 12$
   2. Pronumerals: $a \times b = ab$
   3. Result: $12ab$

   Answer: $12ab$

6. Expanding brackets

   Expand $3(x + 5)$.

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   Key idea: The distributive law says $a(b + c) = ab + ac$. Multiply the term outside the brackets by every term inside.

   1. Multiply $3$ by $x$: $3 \times x = 3x$
   2. Multiply $3$ by $5$: $3 \times 5 = 15$
   3. Write the expanded expression: $3x + 15$

   Answer: $3x + 15$

7. Applying algebra

   A shop charges a flat booking fee of $5 plus $12 per ticket. Write an expression for the total cost $C$ in dollars for $n$ tickets, then find the cost of buying 7 tickets.

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   Key idea: Identify the fixed part and the part that changes with $n$, then substitute.

   1. The cost is a flat $5 plus $12 for each of the $n$ tickets, so $C = 12n + 5$.
   2. Substitute $n = 7$: $C = 12 \times 7 + 5$
   3. Multiply: $C = 84 + 5$
   4. Add: $C = 89$

   Answer: $C = 12n + 5$; the cost for 7 tickets is $89.

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 7 expressions and the distributive law worksheet
• Year 7 algebra worksheet