Grade 8 Algebra: Questions and Step-by-Step Solutions

This page is designed to help students, parents, and teachers master essential Grade 8 algebra concepts through practice questions and step-by-step solutions. Expand the hidden solutions beneath each question to reveal the reasoning behind every step, helping learners build a rock-solid foundation for high school mathematics.

The topics covered on this page include:

Practice Questions

  1. Simplify Algebraic Expressions: Simplify the following expressions.
    • A) \(-2x + 5 + 10x - 9\)
    • B) \(3(x + 7) + 2(-x + 4) + 5x\)
    View Step-by-Step Solution

    A) \(-2x + 5 + 10x - 9\)

    • \(= (10x - 2x) + (5 - 9)\) (Group like terms together)
    • \(= 8x - 4\) (Combine)

    B) \(3(x + 7) + 2(-x + 4) + 5x\)

    • \(= 3x + 21 - 2x + 8 + 5x\) (Expand by distributing the coefficients)
    • \(= (3x - 2x + 5x) + (21 + 8)\) (Group like terms)
    • \(= 6x + 29\) (Combine)
  2. Simplify Rational Expressions:
    • A) \(\dfrac{2x - 6}{2}\)
    • B) \(\dfrac{-x - 2}{x + 2}\)
    • C) \(\dfrac{5x - 5}{10}\)
    View Step-by-Step Solution

    A) \(\dfrac{2x - 6}{2}\)

    • \(= \dfrac{2(x - 3)}{2}\) (Factor out 2 in the numerator)
    • \(= x - 3\) (Divide numerator and denominator by 2 to simplify)

    B) \(\dfrac{-x - 2}{x + 2}\)

    • \(= \dfrac{-1(x + 2)}{x + 2}\) (Factor out -1 in the numerator)
    • \(= -1\) (Cancel out the common term \(x + 2\))

    C) \(\dfrac{5x - 5}{10}\)

    • \(= \dfrac{5(x - 1)}{10}\) (Factor out 5 in the numerator)
    • \(= \dfrac{x - 1}{2}\) (Divide top and bottom by 5)
  3. Solve Equations: Solve for \(x\).
    • A) \(-x = 6\)
    • B) \(2x - 8 = -x + 4\)
    • C) \(2x + \dfrac{1}{2} = \dfrac{2}{3}\)
    • D) \(\dfrac{x}{3} + 2 = 5\)
    • E) \(\dfrac{-5}{x} = 2\)
    View Step-by-Step Solution

    A) \(-x = 6\)

    • \(x = -6\) (Multiply or divide both sides by -1)

    B) \(2x - 8 = -x + 4\)

    • \(2x - 8 + 8 = -x + 4 + 8\) (Add 8 to both sides)
    • \(2x = -x + 12\) (Simplify)
    • \(2x + x = -x + 12 + x\) (Add x to both sides)
    • \(3x = 12\) (Simplify)
    • \(x = 4\) (Multiply both sides by 1/3, or divide by 3)

    C) \(2x + \dfrac{1}{2} = \dfrac{2}{3}\)

    • \(2x = \dfrac{2}{3} - \dfrac{1}{2}\) (Subtract 1/2 from both sides)
    • \(2x = \dfrac{4}{6} - \dfrac{3}{6}\) (Find a common denominator to subtract)
    • \(2x = \dfrac{1}{6}\) (Simplify)
    • \(x = \dfrac{1}{12}\) (Multiply both sides by 1/2)

    D) \(\dfrac{x}{3} + 2 = 5\)

    • \(\dfrac{x}{3} = 3\) (Subtract 2 from both sides)
    • \(x = 9\) (Multiply both sides by 3)

    E) \(\dfrac{-5}{x} = 2\)

    • \(-5 = 2x\) (Multiply both sides by x)
    • \(x = -\dfrac{5}{2}\) (Divide both sides by 2)
  4. Evaluate Expressions: Evaluate for the given values of \(x\) and \(y\).
    • A) \(x^2 - y^2\), for \(x = 4\) and \(y = 5\)
    • B) \(|4x - 2y|\), for \(x = -2\) and \(y = 3\)
    • C) \(3x^3 - 4y^4\), for \(x = -1\) and \(y = -2\)
    View Step-by-Step Solution

    A) \(x^2 - y^2\), for \(x = 4\) and \(y = 5\)

    • \(= 4^2 - 5^2\) (Substitute variables with given values)
    • \(= 16 - 25 = -9\) (Evaluate)

    B) \(|4x - 2y|\), for \(x = -2\) and \(y = 3\)

    • \(= |4(-2) - 2(3)|\) (Substitute values)
    • \(= |-8 - 6| = |-14| = 14\) (Evaluate absolute value)

    C) \(3x^3 - 4y^4\), for \(x = -1\) and \(y = -2\)

    • \(= 3(-1)^3 - 4(-2)^4\) (Substitute values)
    • \(= 3(-1) - 4(16) = -3 - 64 = -67\) (Remember that a negative number to an even power is positive)
  5. Solve Inequalities:
    • A) \(x + 6 < 0\)
    • B) \(x + 1 > 5\)
    • C) \(2(x - 2) < 12\)
    View Step-by-Step Solution

    A) \(x + 6 < 0\)

    • \(x < -6\) (Subtract 6 from both sides)

    B) \(x + 1 > 5\)

    • \(x > 4\) (Subtract 1 from both sides)

    C) \(2(x - 2) < 12\)

    • \(x - 2 < 6\) (Divide both sides by 2)
    • \(x < 8\) (Add 2 to both sides)
  6. Find Reciprocals: What is the reciprocal of each of the following numbers?
    • A) \(-1\)
    • B) \(0\)
    • C) \(\dfrac{3}{4}\)
    • D) \(2\dfrac{5}{7}\)
    • E) \(0.02\)
    View Step-by-Step Solution

    The definition of a reciprocal is: a number \(y\) is the reciprocal of \(x\) if \(x \cdot y = 1\).

    • A) \(-1\): Reciprocal is \(-1\). (\(-1 \cdot -1 = 1\))
    • B) \(0\): Undefined. (No number multiplied by 0 equals 1)
    • C) \(\dfrac{3}{4}\): Reciprocal is \(\dfrac{4}{3}\).
    • D) \(2\dfrac{5}{7}\): First convert to improper fraction \(\dfrac{19}{7}\). The reciprocal is \(\dfrac{7}{19}\).
    • E) \(0.02\): \(0.02 = \dfrac{2}{100} = \dfrac{1}{50}\). The reciprocal is \(50\).
  7. Evaluate Mixed Numbers:
    • A) \(3\dfrac{3}{4} + 6\dfrac{1}{7}\)
    • B) \((1\dfrac{3}{5}) \times (3\dfrac{1}{3}) - 2\dfrac{1}{2}\)
    • C) \((5\dfrac{2}{3}) \div (4\dfrac{1}{5})\)
    • D) \((3\dfrac{4}{7} - 1\dfrac{1}{2}) \div (2\dfrac{3}{8} + 2\dfrac{1}{4})\)
    View Step-by-Step Solution

    A) \(3\dfrac{3}{4} + 6\dfrac{1}{7}\)

    • \(= (3 + 6) + \left(\dfrac{3}{4} + \dfrac{1}{7}\right)\) (Group whole numbers and fractions)
    • \(= 9 + \left(\dfrac{21}{28} + \dfrac{4}{28}\right)\) (Common denominator is 28)
    • \(= 9\dfrac{25}{28}\)

    B) \((1\dfrac{3}{5}) \times (3\dfrac{1}{3}) - 2\dfrac{1}{2}\)

    • \(= \left(\dfrac{8}{5} \times \dfrac{10}{3}\right) - \dfrac{5}{2}\) (Convert to improper fractions)
    • \(= \dfrac{80}{15} - \dfrac{5}{2} = \dfrac{16}{3} - \dfrac{5}{2}\) (Multiply and simplify)
    • \(= \dfrac{32}{6} - \dfrac{15}{6} = \dfrac{17}{6} = 2\dfrac{5}{6}\)

    C) \((5\dfrac{2}{3}) \div (4\dfrac{1}{5})\)

    • \(= \dfrac{17}{3} \div \dfrac{21}{5}\) (Convert to improper fractions)
    • \(= \dfrac{17}{3} \times \dfrac{5}{21} = \dfrac{85}{63} = 1\dfrac{22}{63}\) (Multiply by the reciprocal)

    D) \((3\dfrac{4}{7} - 1\dfrac{1}{2}) \div (2\dfrac{3}{8} + 2\dfrac{1}{4})\)

    • \(= \left(\dfrac{25}{7} - \dfrac{3}{2}\right) \div \left(\dfrac{19}{8} + \dfrac{9}{4}\right)\) (Convert to improper fractions)
    • \(= \left(\dfrac{50}{14} - \dfrac{21}{14}\right) \div \left(\dfrac{19}{8} + \dfrac{18}{8}\right)\) (Find common denominators)
    • \(= \dfrac{29}{14} \div \dfrac{37}{8} = \dfrac{29}{14} \times \dfrac{8}{37} = \dfrac{232}{518} = \dfrac{116}{259}\) (Multiply by reciprocal and reduce)
  8. Evaluate Exponential Expressions:
    • A) \(-4^2\)
    • B) \((-2)^3\)
    • C) \((-2)^4\)
    • D) \(1000^0\)
    • E) \(566^1\)
    View Step-by-Step Solution
    • A) \(-4^2\) = \(-(4 \times 4)\) = \(-16\) (The negative sign is outside the base)
    • B) \((-2)^3\) = \((-2) \times (-2) \times (-2)\) = \(-8\)
    • C) \((-2)^4\) = \((-2) \times (-2) \times (-2) \times (-2)\) = \(16\) (A negative base to an even power is positive)
    • D) \(1000^0\) = \(1\) (Any non-zero number to the power of 0 equals 1)
    • E) \(566^1\) = \(566\) (Any number to the power of 1 is the number itself)
  9. Convert to Fractions: Write in simplest fraction form.
    • A) \(0.02\)
    • B) \(12\%\)
    • C) \(0.5\%\)
    • D) \(1.12\)
    View Step-by-Step Solution
    • A) \(0.02\) = \(\dfrac{2}{100}\) = \(\dfrac{1}{50}\)
    • B) \(12\%\) = \(\dfrac{12}{100}\) = \(\dfrac{3}{25}\)
    • C) \(0.5\%\) = \(\dfrac{0.5}{100}\) = \(\dfrac{5}{1000}\) = \(\dfrac{1}{200}\)
    • D) \(1.12\) = \(\dfrac{112}{100}\) = \(\dfrac{28}{25}\)
  10. Convert to Decimals:
    • A) \(\dfrac{1}{5}\)
    • B) \(120\%\)
    • C) \(0.2\%\)
    • D) \(4\dfrac{8}{5}\)
    View Step-by-Step Solution
    • A) \(\dfrac{1}{5}\) = \(0.2\)
    • B) \(120\%\) = \(\dfrac{120}{100}\) = \(1.2\)
    • C) \(0.2\%\) = \(\dfrac{0.2}{100}\) = \(0.002\)
    • D) \(4\dfrac{8}{5}\) = \(4 + 1.6\) = \(5.6\) (Note: 8/5 is improper, equaling 1.6)
  11. Convert to Percents:
    • A) \(\dfrac{3}{10}\)
    • B) \(1.4\)
    • C) \(123.45\)
    • D) \(2\dfrac{4}{5}\)
    View Step-by-Step Solution
    • A) \(\dfrac{3}{10}\) = \(\dfrac{30}{100}\) = \(30\%\)
    • B) \(1.4\) = \(\dfrac{140}{100}\) = \(140\%\)
    • C) \(123.45\) = \(\dfrac{12345}{100}\) = \(12345\%\)
    • D) \(2\dfrac{4}{5}\) = \(2.8\) = \(\dfrac{280}{100}\) = \(280\%\)
  12. Divisibility by 3: Which of these numbers is divisible by 3? (A) 156,312 (B) 176,314
    View Step-by-Step Solution

    A number is divisible by 3 if the sum of its digits is divisible by 3.

    • A) 156,312: \(1+5+6+3+1+2 = 18\). 18 is divisible by 3, so YES.
    • B) 176,314: \(1+7+6+3+1+4 = 22\). 22 is not divisible by 3, so NO.
  13. Divisibility by 4: Which of these numbers is divisible by 4? (A) 3,432 (B) 1,257
    View Step-by-Step Solution

    A number is divisible by 4 if its last two digits form a number divisible by 4.

    • A) 3,432: The last two digits are 32. 32 is divisible by 4, so YES.
    • B) 1,257: The last two digits are 57. 57 is odd and not divisible by 4, so NO.
  14. Divisibility by 6: Which of these numbers is divisible by 6? (A) 1,233 (B) 3,432
    View Step-by-Step Solution

    A number is divisible by 6 if it is divisible by BOTH 2 and 3.

    • A) 1,233: Ends in an odd number, so it is NOT divisible by 2. Therefore, NO.
    • B) 3,432: Sum of digits is 12 (divisible by 3) and it ends in 2 (even/divisible by 2). Therefore, YES.
  15. Divisibility by 9: Which of these numbers is divisible by 9? (A) 2,538 (B) 1,451
    View Step-by-Step Solution

    A number is divisible by 9 if the sum of its digits is divisible by 9.

    • A) 2,538: \(2+5+3+8 = 18\). 18 is divisible by 9, so YES.
    • B) 1,451: \(1+4+5+1 = 11\). 11 is not divisible by 9, so NO.
  16. Algebraic Substitution: Evaluate \(8x + 7\) given that \(x - 3 = 10\).
    View Step-by-Step Solution
    • \(x - 3 = 10\) (Given equation)
    • \(x = 10 + 3 = 13\) (Solve for x)
    • \(8(13) + 7\) (Substitute x by 13 into the expression)
    • \(= 104 + 7 = 111\) (Evaluate)

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