About this preparation
Math diagnostic test, Grade 9: Linear equations, Systems of equations. 10 questions with answers and explanations. Topics: Linear equations, Systems of equations, Quadratic expressions, Factorization, Pythagorean theorem, Percentages, Ratios and proportions, Basic geometry. Duration: 55 min.
Diagnostic tasks
Below is the full list of tasks with correct answers and explanations. In a lesson the tutor can hide the answers and reveal them after the student tries.
1
Solve the equation: 3x - 7 = 11.
x = 4
x = 6
x = 7
x = 8
Explanation: Add 7 to both sides to get 3x = 18, then divide by 3. So x = 6.
2
Which expression is equivalent to 2(x + 5) - 3x?
x + 10
-x + 10
x - 10
-x - 10
Explanation: Distribute the 2: 2x + 10 - 3x = -x + 10. The correct equivalent expression is -x + 10.
3
Solve the system of equations: x + y = 10 and x - y = 2.
x = 4, y = 6
x = 6, y = 4
x = 5, y = 5
x = 8, y = 2
Explanation: Add the equations: 2x = 12, so x = 6. Substitute into x + y = 10 to get y = 4.
4
Factor the expression: x² - 9.
(x - 3)(x - 3)
(x - 9)(x + 1)
(x - 3)(x + 3)
(x + 9)(x - 1)
Explanation: This is a difference of squares: a² - b² = (a - b)(a + b). So x² - 9 = x² - 3² = (x - 3)(x + 3).
5
Which of the following are true statements about a right triangle with legs 6 and 8?
The hypotenuse is 10
The area is 24
The perimeter is 20
The triangle is isosceles
Explanation: Using the Pythagorean theorem, the hypotenuse is √(6² + 8²) = √100 = 10. The area is 1/2 × 6 × 8 = 24. The perimeter is 6 + 8 + 10 = 24, so 20 is false.
6
A jacket costs $80 and is discounted by 15%. What is the sale price?
$68
$72
$75
$85
Explanation: 15% of 80 is 12. Subtract 12 from 80 to get 68.
7
If 5 notebooks cost $12.50, how much do 8 notebooks cost at the same rate?
$18.00
$19.50
$20.00
$22.50
Explanation: Find the unit price: 12.50 ÷ 5 = 2.50 per notebook. Then 8 × 2.50 = 20.00.
8
Which value of x makes the equation 4x + 1 = 2x + 9 true?
x = 2
x = 3
x = 4
x = 5
Explanation: Subtract 2x from both sides: 2x + 1 = 9. Then subtract 1: 2x = 8, so x = 4? Wait carefully: 4x + 1 = 2x + 9 gives 2x + 1 = 9, then 2x = 8, so x = 4.
9
Which of the following are solutions of the inequality x > 3?
4
3
7
2
Explanation: The inequality x > 3 means any number greater than 3 is a solution. Here, 4 and 7 work, while 3 and 2 do not.
10
A rectangle has length 12 cm and width 5 cm. What is its diagonal length?
13 cm
17 cm
7 cm
60 cm
Explanation: Use the Pythagorean theorem on the sides 12 and 5: diagonal = √(12² + 5²) = √169 = 13 cm.
