Unit notes and practice questions

Grade 9 Mathematics unit 3: Solving Equations

Everything here is for unit 3 of the Grade 9 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

5
Cards: 10
Practice questions on unit 3: 87–129
The textbook pages for unit 3

What unit 3 covers

1.Revision of Linear Equation in One Variable
2.Systems of Linear Equations in Two Variables
3.Solving Non-linear Equations
4.Some Applications of Solving Equations

Short notes for unit 3

The cards filed under this chapter on the Grade 9 Mathematics page. Open one at a time.

Absolute Value Equation RuleRule

When you use it

Use when solving an equation where an algebraic expression is enclosed inside absolute value bars.

Watch out

If the isolated absolute value equals a negative number, the equation has no solution immediately.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Discriminant and Nature of RootsReference tableD = b^2 - 4ac > 0 · D = b^2 - 4ac = 0 · D = b^2 - 4ac < 0

Discriminant Value	Nature of Roots	Number of Real Roots
$D = b^2 - 4ac > 0$	Real and distinct	$2$
$D = b^2 - 4ac = 0$	Real and equal	$1$
$D = b^2 - 4ac < 0$	No real roots	$0$

When you use it

Use to determine the number and type of roots of a quadratic equation without solving it completely.

Watch out

A negative discriminant means there are no real roots. Do not take the square root of a negative number in real numbers.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Extraneous Roots in Radical EquationsCommon mistake

When you use it

Use when solving equations containing variables under square roots or fractional exponents.

Watch out

Squaring both sides can introduce extraneous solutions. You must check every candidate root in the original equation.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Quadratic FormulaFormula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

$x$: roots of the quadratic equation $none$
$a$: leading coefficient where a is not zero $none$
$b$: coefficient of the middle term $none$
$c$: constant term $none$

When you use it

Use to find the solutions of any quadratic equation in the standard form ax^2 + bx + c = 0.

Watch out

The division by 2a applies to the entire numerator including -b, not just the square root part.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Solvability of Linear SystemsReference tablea_1/a_2 is not equal to b_1/b_2 · a_1/a_2 = b_1/b_2 = c_1/c_2 · a_1/a_2 = b_1/b_2 is not equal to c_1/c_2

Ratio Condition	System Classification	Number of Solutions	Geometric Lines
$\frac{a_1}{a_2}$ is not equal to $\frac{b_1}{b_2}$	Independent	One unique solution	Intersecting
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Dependent	Infinitely many	Coincident
$\frac{a_1}{a_2} = \frac{b_1}{b_2}$ is not equal to $\frac{c_1}{c_2}$	Inconsistent	No solution	Parallel

When you use it

Use to classify a two-variable linear system as independent, dependent, or inconsistent using coefficient ratios.

Watch out

Make sure equations are aligned in the same standard form before comparing coefficients.

Drafted from Grade 9 Mathematics, pages 87-129, then checked twice before it went up

Practice questions on unit 3

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

Solve the linear equation 3x + 4 = 10 for x.

A.x = 2
B.x = 3
C.x = 6
D.x = 14

Show the answer and why

Answer: A x = 2

Why

Subtract 4 from both sides: 3x = 10 - 4 = 6. Then divide both sides by 3 to get x = 2.

Question 2

What are the solutions to the absolute value equation |x| = 4?

A.x = 4 only
B.x = -4 only
C.x = 4 or x = -4
D.No real solution

Show the answer and why

Answer: C x = 4 or x = -4

Why

By the definition of absolute value, for any non-negative number p, |x| = p means x = p or x = -p. Therefore, x = 4 or x = -4.

Question 3

Solve the exponential equation 2^x+4 = 32.

A.x = 1
B.x = 2
C.x = 5
D.x = 9

Show the answer and why

Answer: A x = 1

Why

Write 32 with base 2: 32 = 2⁵. Since bases are equal, equate the exponents: x + 4 = 5, which gives x = 1.

Question 4

For a quadratic equation with real coefficients, what is true about the roots if the discriminant D = b² - 4ac > 0?

A.There are no real roots
B.The roots are real and distinct
C.The roots are real and equal
D.The equation has infinitely many solutions

Show the answer and why

Answer: B The roots are real and distinct

Why

When the discriminant D > 0, the quadratic equation has two distinct (unequal) real roots.

Question 5

What is the solution (x, y) to the following system of linear equations?
x + 2y = 1
-x + y = 5

A.(3, -4)
B.(-3, 2)
C.(2, -3)
D.(-1, 1)

Show the answer and why

Answer: B (-3, 2)

Why

Add the two equations together: (x + -x) + (2y + y) = 1 + 5, which gives 3y = 6, so y = 2. Substituting into -x + 2 = 5 gives x = -3. Thus, the solution is (-3, 2).

Question 6

What are the solutions to the equation |x + 7| = 14?

A.x = 7 or x = -21
B.x = 7 or x = -7
C.x = 14 or x = -14
D.x = 21 or x = -7

Show the answer and why

Answer: A x = 7 or x = -21

Why

Set up two linear equations: x + 7 = 14 or x + 7 = -14. Solving these gives x = 14 - 7 = 7 or x = -14 - 7 = -21.

Question 7

Solve the quadratic equation x² - 6x - 16 = 0 by factorization.

A.x = -8 or x = 2
B.x = 8 or x = -2
C.x = 4 or x = -4
D.x = -16 or x = 1

Show the answer and why

Answer: B x = 8 or x = -2

Why

Find two numbers whose sum is -6 and product is -16. These are -8 and 2. Factorizing gives (x - 8)(x + 2) = 0, so x = 8 or x = -2.

Question 8

What is the solution to the radical equation √(15 - 2x) = x?

A.x = 3 and x = -5
B.x = -5 only
C.x = 3 only
D.No real solution

Show the answer and why

Answer: C x = 3 only

Why

Squaring both sides gives 15 - 2x = x², or x² + 2x - 15 = 0, which factors into (x - 3)(x + 5) = 0. Since x ≥ 0 for the square root, x = -5 cannot be a solution. The only solution is x = 3.

Question 9

Tickets to a circus in Addis Ababa cost Birr 25 for children and Birr 50 for adults. On one day, total attendance was 2,000 people and total revenue was Birr 70,000. How many children bought tickets?

A.800
B.1,000
C.1,200
D.1,500

Show the answer and why

Answer: C 1,200

Why

Let c be children and a be adults. The system is c + a = 2000 and 25c + 50a = 70000. Substituting a = 2000 - c into the second equation gives 25c + 50(2000 - c) = 70000, which simplifies to -25c = -30000, so c = 1200.

Question 10

If the difference between the roots of the quadratic equation x² - 13x + k = 0 is 7, what is the value of k?

A.30
B.20
C.13
D.40

Show the answer and why

Answer: A 30

Why

Let the roots be r₁ and r₂. From the coefficients, r₁ + r₂ = 13. Given r₁ - r₂ = 7, solving the two equations gives r₁ = 10 and r₂ = 3. Therefore, k = r₁ × r₂ = 10 × 3 = 30.

The app holds 12 questions on this unit, marked as you go. These are the first 10.