Grade 9 Algebra: Practice Questions and Solutions

This page provides a comprehensive set of Grade 9 algebra practice questions complete with step-by-step solutions. Expand the hidden solutions beneath each question block to reveal detailed algebraic reasoning.

Topics covered in this practice set include:

Algebra Practice Problems

  1. Simplifying Algebraic Expressions: Simplify the following algebraic expressions.
    1. \( -6x + 5 + 12x - 6 \)
    2. \( 2(x - 9) + 6(-x + 2) + 4x \)
    3. \( 3x^2 + 12 + 9x - 20 + 6x^2 - x \)
    4. \( (x + 2)(x + 4) + (x + 5)(-x - 1) \)
    5. \( 1.2(x - 9) - 2.3(x + 4) \)
    6. \( (x^2y)(xy^2) \)
    7. \( (-x^2y^2)(xy^2) \)
    View Step-by-Step Solutions

    1. Group like terms and simplify:

      • \( (-6x + 12x) + (5 - 6) \)
      • \( = 6x - 1 \)

    2. Expand brackets, then group like terms:

      • \( 2x - 18 - 6x + 12 + 4x \)
      • \( (2x - 6x + 4x) + (-18 + 12) \)
      • \( = -6 \)

    3. Group like terms and simplify:

      • \( (3x^2 + 6x^2) + (9x - x) + (12 - 20) \)
      • \( = 9x^2 + 8x - 8 \)

    4. Expand brackets and collect like terms:

      • \( (x^2 + 4x + 2x + 8) + (-x^2 - x - 5x - 5) \)
      • \( x^2 + 6x + 8 - x^2 - 6x - 5 \)
      • \( (x^2 - x^2) + (6x - 6x) + (8 - 5) \)
      • \( = 3 \)

    5. Expand and group decimals:

      • \( 1.2x - 10.8 - 2.3x - 9.2 \)
      • \( (1.2x - 2.3x) + (-10.8 - 9.2) \)
      • \( = -1.1x - 20 \)

    6. Rewrite using exponential rules (add exponents of like bases):

      • \( (x^2 \cdot x^1)(y^1 \cdot y^2) \)
      • \( = x^3y^3 \)

    7. Apply exponential rules and retain the negative sign:

      • \( -(x^2 \cdot x^1)(y^2 \cdot y^2) \)
      • \( = -x^3y^4 \)
  2. Rational Expressions: Simplify the following algebraic fractions.
    1. \( \dfrac{(a b^2)(a^3 b)}{a^2 b^3} \)
    2. \( \dfrac{21 x^5}{3 x^4} \)
    3. \( \dfrac{(6 x^4)(4 y^2)}{(3 x^2)(16 y)} \)
    4. \( \dfrac{4x - 12}{4} \)
    5. \( \dfrac{-5x - 10}{x + 2} \)
    6. \( \dfrac{x^2 - 4x - 12}{x^2 - 2x - 24} \)
    View Step-by-Step Solutions

    1. Simplify the numerator first, then divide:

      • Numerator: \( a^1 \cdot a^3 = a^4 \), and \( b^2 \cdot b^1 = b^3 \) \(\rightarrow a^4b^3\)
      • Fraction: \( \dfrac{a^4 b^3}{a^2 b^3} = \left(\dfrac{a^4}{a^2}\right) \cdot \left(\dfrac{b^3}{b^3}\right) \)
      • \( = a^2 \)

    2. Divide coefficients and apply exponent quotient rule:

      • \( \left(\dfrac{21}{3}\right) \cdot \left(\dfrac{x^5}{x^4}\right) \)
      • \( = 7x \)

    3. Multiply terms in the numerator and denominator, then simplify:

      • \( \dfrac{24 x^4 y^2}{48 x^2 y} \)
      • \( \left(\dfrac{24}{48}\right) \cdot \left(\dfrac{x^4}{x^2}\right) \cdot \left(\dfrac{y^2}{y^1}\right) \)
      • \( = \dfrac{1}{2} x^2 y \)

    4. Factor the common term from the numerator:

      • \( \dfrac{4(x - 3)}{4} \)
      • \( = x - 3 \)

    5. Factor -5 out of the numerator:

      • \( \dfrac{-5(x + 2)}{x + 2} \)
      • Cancel the common factor \( (x + 2) \) (assuming \(x \neq -2\))
      • \( = -5 \)

    6. Factor both the numerator and denominator:

      • Numerator: \( (x - 6)(x + 2) \)
      • Denominator: \( (x - 6)(x + 4) \)
      • \( \dfrac{(x - 6)(x + 2)}{(x - 6)(x + 4)} \)
      • \( = \dfrac{x + 2}{x + 4} \) (for \( x \neq 6 \) and \( x \neq -4 \))
  3. Linear Equations: Solve for \( x \) in the following equations.
    1. \( 2x = 6 \)
    2. \( 6x - 8 = 4x + 4 \)
    3. \( 4(x - 2) = 2(x + 3) + 7 \)
    4. \( 0.1x - 1.6 = 0.2x + 2.3 \)
    5. \( -\dfrac{x}{5} = 2 \)
    6. \( \dfrac{x - 4}{-6} = 3 \)
    7. \( \dfrac{-3x + 1}{x - 2} = -3 \)
    8. \( \dfrac{x}{5} + \dfrac{x - 1}{3} = \dfrac{1}{5} \)
    View Step-by-Step Solutions
      • Divide both sides by 2: \( x = \dfrac{6}{2} \)
      • \( x = 3 \)
      • Subtract \(4x\) and add \(8\) to both sides:
      • \( 6x - 4x = 4 + 8 \)
      • \( 2x = 12 \)
      • \( x = 6 \)
      • Expand the brackets: \( 4x - 8 = 2x + 6 + 7 \)
      • Simplify right side: \( 4x - 8 = 2x + 13 \)
      • Subtract \(2x\) and add \(8\): \( 4x - 2x = 13 + 8 \)
      • \( 2x = 21 \)
      • \( x = \dfrac{21}{2} \)
      • Subtract \(0.2x\) and add \(1.6\):
      • \( 0.1x - 0.2x = 2.3 + 1.6 \)
      • \( -0.1x = 3.9 \)
      • Divide by -0.1: \( x = \dfrac{3.9}{-0.1} \)
      • \( x = -39 \)
      • Multiply both sides by -5:
      • \( x = -10 \)
      • Multiply both sides by -6: \( x - 4 = -18 \)
      • Add 4: \( x = -18 + 4 \)
      • \( x = -14 \)
      • Multiply both sides by \( (x - 2) \):
      • \( -3x + 1 = -3(x - 2) \)
      • Expand right side: \( -3x + 1 = -3x + 6 \)
      • Add \(3x\) to both sides: \( 1 = 6 \)
      • This is a false statement. No real solution.
      • Multiply all terms by the Least Common Multiple (15):
      • \( 15\left(\dfrac{x}{5}\right) + 15\left(\dfrac{x - 1}{3}\right) = 15\left(\dfrac{1}{5}\right) \)
      • Simplify: \( 3x + 5(x - 1) = 3 \)
      • Expand: \( 3x + 5x - 5 = 3 \)
      • Combine like terms: \( 8x - 5 = 3 \)
      • Add 5 and divide by 8: \( 8x = 8 \)
      • \( x = 1 \)
  4. Quadratic Equations: Find all real solutions for the following equations.
    1. \( 2x^2 - 8 = 0 \)
    2. \( x^2 = -5 \)
    3. \( 2x^2 + 5x - 7 = 0 \)
    4. \( (x - 2)(x + 3) = 0 \)
    5. \( (x + 7)(x - 1) = 9 \)
    6. \( x(x - 6) = -9 \)
    View Step-by-Step Solutions
      • Divide by 2: \( x^2 - 4 = 0 \)
      • Factor difference of squares: \( (x - 2)(x + 2) = 0 \)
      • Solution set: \( \{-2, 2\} \)
      • The square of a real number cannot be negative.
      • No real solution.
      • Factor the left side: \( (2x + 7)(x - 1) = 0 \)
      • Set factors to zero: \( 2x + 7 = 0 \Rightarrow x = -\dfrac{7}{2} \)
      • \( x - 1 = 0 \Rightarrow x = 1 \)
      • Solution set: \( \left\{-\dfrac{7}{2}, 1\right\} \)
      • The equation is already factored. Set factors to zero:
      • \( x - 2 = 0 \Rightarrow x = 2 \)
      • \( x + 3 = 0 \Rightarrow x = -3 \)
      • Solution set: \( \{-3, 2\} \)
      • Expand the left side: \( x^2 + 6x - 7 = 9 \)
      • Set equal to zero: \( x^2 + 6x - 16 = 0 \)
      • Factor: \( (x + 8)(x - 2) = 0 \)
      • Solution set: \( \{-8, 2\} \)
      • Expand: \( x^2 - 6x = -9 \)
      • Set equal to zero: \( x^2 - 6x + 9 = 0 \)
      • Factor perfect square: \( (x - 3)^2 = 0 \)
      • Solution set: \( \{3\} \)
  5. Other Non-Linear Equations: Find any real solutions for the following equations.
    1. \( x^3 - 1728 = 0 \)
    2. \( x^3 = -64 \)
    3. \( \sqrt{x} = -1 \)
    4. \( \sqrt{x} = 5 \)
    5. \( \sqrt{\dfrac{x}{100}} = 4 \)
    6. \( \sqrt{\dfrac{200}{x}} = 2 \)
    View Step-by-Step Solutions
      • Rewrite: \( x^3 = 1728 \)
      • Take the cube root: \( x = \sqrt[3]{1728} \)
      • \( x = 12 \)
      • Take the cube root: \( x = \sqrt[3]{-64} \)
      • \( x = -4 \)
      • The principal square root of a real number is never negative.
      • No real solution.
      • Square both sides: \( (\sqrt{x})^2 = 5^2 \)
      • \( x = 25 \)
      • Square both sides: \( \dfrac{x}{100} = 16 \)
      • Multiply by 100: \( x = 16 \times 100 \)
      • \( x = 1600 \)
      • Square both sides: \( \dfrac{200}{x} = 4 \)
      • Multiply by \( x \) and divide by 4: \( x = \dfrac{200}{4} \)
      • \( x = 50 \)
  6. Evaluating Expressions: Evaluate for the given values of \( a \) and \( b \).
    1. \( a^2 + b^2 \) for \( a = 2 \) and \( b = 2 \)
    2. \( |2a - 3b| \) for \( a = -3 \) and \( b = 5 \)
    3. \( 3a^3 - 4b^4 \) for \( a = -1 \) and \( b = -2 \)
    View Step-by-Step Solutions
      • Substitute values: \( (2)^2 + (2)^2 \)
      • \( = 4 + 4 \)
      • \( = 8 \)
      • Substitute values: \( |2(-3) - 3(5)| \)
      • \( = |-6 - 15| \)
      • \( = |-21| \)
      • \( = 21 \) (Absolute value removes the negative)
      • Substitute values: \( 3(-1)^3 - 4(-2)^4 \)
      • Evaluate exponents first: \( 3(-1) - 4(16) \)
      • \( = -3 - 64 \)
      • \( = -67 \)
  7. Inequalities: Solve the following inequalities.
    1. \( x + 3 < 0 \)
    2. \( x + 1 > -x + 5 \)
    3. \( 2(x - 2) < -(x + 7) \)
    View Step-by-Step Solutions
      • Subtract 3 from both sides:
      • \( x < -3 \)
      • Add \( x \) to both sides: \( 2x + 1 > 5 \)
      • Subtract 1: \( 2x > 4 \)
      • Divide by 2: \( x > 2 \)
      • Expand both sides: \( 2x - 4 < -x - 7 \)
      • Add \( x \): \( 3x - 4 < -7 \)
      • Add 4: \( 3x < -3 \)
      • Divide by 3: \( x < -1 \)
  8. Discriminant: For what value of the constant \( k \) does the quadratic equation \( x^2 + 2x = -2k \) have two distinct real solutions?
    View Step-by-Step Solution
    • First, write the equation in standard form (\(ax^2 + bx + c = 0\)):
      \( x^2 + 2x + 2k = 0 \)
    • Identify coefficients: \( a = 1, b = 2, c = 2k \)
    • Calculate the discriminant \( D = b^2 - 4ac \):
      \( D = (2)^2 - 4(1)(2k) = 4 - 8k \)
    • For two distinct real solutions, the discriminant must be strictly positive (\( D > 0 \)):
      \( 4 - 8k > 0 \)
    • Solve the inequality: \( 4 > 8k \)
    • \( k < \dfrac{1}{2} \)
  9. Linear Equation Slopes: For what value of the constant \( b \) does the linear equation \( 2x + by = 2 \) have a slope equal to 2?
    View Step-by-Step Solution
    • Solve for \( y \) to find the slope-intercept form (\( y = mx + c \)):
    • \( by = -2x + 2 \)
    • \( y = -\dfrac{2}{b}x + \dfrac{2}{b} \)
    • The slope \( m \) is \( -\dfrac{2}{b} \). Set this equal to 2:
    • \( -\dfrac{2}{b} = 2 \)
    • Multiply by \( b \): \( -2 = 2b \)
    • \( b = -1 \)
  10. Intercepts: What is the y-intercept of the line \( -4x + 6y = -12 \)?
    View Step-by-Step Solution
    • To find the y-intercept, set \( x = 0 \) and solve for \( y \):
    • \( -4(0) + 6y = -12 \)
    • \( 6y = -12 \)
    • \( y = -2 \)
    • y-intercept: \( (0, -2) \)
  11. Intercepts: What is the x-intercept of the line \( -3x + y = 3 \)?
    View Step-by-Step Solution
    • To find the x-intercept, set \( y = 0 \) and solve for \( x \):
    • \( -3x + 0 = 3 \)
    • \( -3x = 3 \)
    • \( x = -1 \)
    • x-intercept: \( (-1, 0) \)
  12. Systems of Linear Equations: What is the point of intersection of the lines \( x - y = 3 \) and \( -5x - 2y = -22 \)?
    View Step-by-Step Solution
    • A point of intersection solves both equations simultaneously. Use substitution.
    • Solve the first equation for \( x \):
      \( x = 3 + y \)
    • Substitute \( (3 + y) \) for \( x \) in the second equation:
      \( -5(3 + y) - 2y = -22 \)
    • Expand and simplify:
      \( -15 - 5y - 2y = -22 \)
      \( -7y - 15 = -22 \)
    • Add 15 to both sides:
      \( -7y = -7 \)
      \( y = 1 \)
    • Substitute \( y = 1 \) back into \( x = 3 + y \):
      \( x = 3 + 1 = 4 \)
    • Point of intersection: \( (4, 1) \)
  13. Points on a Line: For what value of the constant \( k \) does the line \( -4x + ky = 2 \) pass through the point \( (2, -3) \)?
    View Step-by-Step Solution
    • If a line passes through a point, those coordinates satisfy the equation. Substitute \( x = 2 \) and \( y = -3 \):
    • \( -4(2) + k(-3) = 2 \)
    • \( -8 - 3k = 2 \)
    • Add 8 to both sides: \( -3k = 10 \)
    • \( k = -\dfrac{10}{3} \)
  14. Slopes of Special Lines: What is the slope of the line with equation \( y - 4 = 10 \)?
    View Step-by-Step Solution
    • Simplify the equation to slope-intercept form (\( y = mx + c \)):
    • \( y = 14 \)
    • This represents a horizontal line. Since the \( x \)-term is missing, \( m = 0 \).
    • Slope = 0
  15. Slopes of Special Lines: What is the slope of the line with equation \( 2x = -8 \)?
    View Step-by-Step Solution
    • Simplify the equation:
    • \( x = -4 \)
    • This represents a vertical line. Vertical lines do not have a defined rise-over-run because the horizontal change is zero (division by zero).
    • Slope is undefined
  16. Intercepts of Special Lines: Find the \( x \) and \( y \) intercepts of the line with equation \( x = -3 \).
    View Step-by-Step Solution
    • \( x = -3 \) is a vertical line running parallel to the y-axis.
    • It crosses the x-axis at -3, so the x-intercept is \( (-3, 0) \).
    • Because it is parallel to the y-axis, it will never cross it. It has no y-intercept.
  17. Intercepts of Special Lines: Find the \( x \) and \( y \) intercepts of the line with equation \( 3y - 6 = 3 \).
    View Step-by-Step Solution
    • Simplify the equation:
      \( 3y = 9 \Rightarrow y = 3 \)
    • \( y = 3 \) is a horizontal line running parallel to the x-axis.
    • It crosses the y-axis at 3, so the y-intercept is \( (0, 3) \).
    • Because it is parallel to the x-axis, it will never cross it. It has no x-intercept.
  18. Slopes and Axes: What is the slope of a line parallel to the x-axis?
    View Step-by-Step Solution
    • A line parallel to the x-axis is a horizontal line.
    • A horizontal line does not rise or fall, therefore its slope is 0.
  19. Slopes and Axes: What is the slope of a line perpendicular to the x-axis?
    View Step-by-Step Solution
    • A line perpendicular to the x-axis is a vertical line (parallel to the y-axis).
    • The change in \( x \) is zero, meaning you would divide by zero to calculate slope.
    • The slope is undefined.

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