Year 9 Expressions and linear equations: free practice test with answers

Sample expressions and linear equations questions with answers

Try each one before you open the answer.

1. Algebraic expressions

   Write an algebraic expression for each of the following.

   (a) $7$ more than the product of $3$ and $x$

   (b) the sum of $a$ and $b$, divided by $4$

   (c) half of the square of $m$

   Show the answer

   Key idea: Translate each phrase in order: "product" means multiply, "sum" means add, and "square" means raise to the power $2$.

   1. (a) The product of $3$ and $x$ is $3x$; $7$ more than this is $3x + 7$
   2. (b) The sum of $a$ and $b$ is $a + b$; divided by $4$ gives $\frac{a + b}{4}$
   3. (c) The square of $m$ is $m^2$; half of it is $\frac{m^2}{2}$

   Answer: (a) $3x + 7$ (b) $\frac{a + b}{4}$ (c) $\frac{m^2}{2}$

2. Simplifying algebraic expressions

   Simplify:

   (a) $4ab - 7ab + 2ab$

   (b) $3x^2 + 5x - x^2 - 8x$

   Show the answer

   Key idea: Only like terms (same pronumerals with the same powers) can be added or subtracted.

   1. (a) $4 - 7 + 2 = -1$, so $4ab - 7ab + 2ab = -ab$
   2. (b) $x^2$ terms: $3x^2 - x^2 = 2x^2$; $x$ terms: $5x - 8x = -3x$, giving $2x^2 - 3x$

   Answer: (a) $-ab$ (b) $2x^2 - 3x$

3. Expanding algebraic expressions

   Expand and simplify:

   (a) $-2x(3x - 5)$

   (b) $4(2a - 3) - 5(a - 2)$

   Show the answer

   Key idea: Multiply every term inside the brackets by the term outside, taking care with negative signs, then collect like terms.

   1. (a) $-2x \times 3x = -6x^2$ and $-2x \times (-5) = 10x$, so $-6x^2 + 10x$
   2. (b) $4(2a - 3) = 8a - 12$ and $-5(a - 2) = -5a + 10$
   3. (b) $8a - 12 - 5a + 10 = 3a - 2$

   Answer: (a) $-6x^2 + 10x$ (b) $3a - 2$

4. Solving equations with brackets and pronumerals on both sides

   Solve $3(2x - 4) = 2x + 8$.

   Show the answer

   Key idea: Expand brackets first, then collect like terms and isolate $x$.

   1. Expand the left side: $6x - 12 = 2x + 8$.
   2. Subtract $2x$ from both sides: $4x - 12 = 8$.
   3. Add $12$ to both sides: $4x = 20$.
   4. Divide both sides by $4$: $x = 5$.
   5. Check: $3(2(5) - 4) = 3(6) = 18$ and $2(5) + 8 = 18$. ✓

   Answer: $x = 5$

5. Solving word problems

   The sum of three consecutive whole numbers is $87$. Write and solve an equation to find the three numbers.

   Show the answer

   Key idea: Let the smallest number be $n$. Consecutive numbers are $n$, $n + 1$ and $n + 2$.

   1. $n + (n + 1) + (n + 2) = 87$
   2. $3n + 3 = 87$, so $3n = 84$ and $n = 28$
   3. The numbers are $28$, $29$ and $30$ (check: $28 + 29 + 30 = 87$)

   Answer: $28$, $29$ and $30$

6. Solving linear inequalities

   Solve each inequality.

   (a) $3x + 4 > 19$

   (b) $2x - 7 \le 9$

   Show the answer

   Key idea: Solve an inequality like an equation. The inequality sign stays the same when you add, subtract, or multiply or divide by a positive number.

   1. (a) Subtract $4$: $3x > 15$; divide by $3$: $x > 5$
   2. (b) Add $7$: $2x \le 16$; divide by $2$: $x \le 8$

   Answer: (a) $x > 5$ (b) $x \le 8$

7. Simultaneous equations using substitution

   Use substitution to solve the simultaneous equations $y = 2x + 1$ and $x + y = 10$.

   Show the answer

   Key idea: Replace $y$ in the second equation with the expression it equals from the first equation.

   1. $x + (2x + 1) = 10$
   2. $3x + 1 = 10$, so $x = 3$
   3. $y = 2(3) + 1 = 7$ (check: $3 + 7 = 10$ ✓)

   Answer: $x = 3$, $y = 7$

8. Simultaneous equations using elimination

   Use elimination to solve $x + y = 9$ and $x - y = 3$.

   Show the answer

   Key idea: The $y$ terms have opposite signs, so adding the equations eliminates $y$.

   1. Add: $2x = 12$, so $x = 6$
   2. Substitute into $x + y = 9$: $6 + y = 9$, so $y = 3$

   Answer: $x = 6$, $y = 3$

Free worksheets for this chapter

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

• Year 9 algebra worksheet
• Year 9 linear equations and graphs worksheet
• Year 9 simple interest worksheet