Algebraic Expressions: Questions and Step-by-Step Solutions - Grade 6

This page is designed to help students, parents, and teachers master the topic of algebraic expressions through carefully selected questions and step-by-step solutions with explanations. Each solution goes beyond providing the final answer by explaining the reasoning behind every step, helping learners build a strong foundation in algebra. Since algebraic expressions are one of the most essential concepts in understanding algebra, it is crucial to study them thoroughly to succeed in more advanced mathematics.

The questions on this page cover essential topics related to algebraic expressions, including:

Questions

  1. Identify Algebraic Expressions: Which of the following is an algebraic expression?
    • \(2x + 2\)
    • \(x + 2 \& \; 8\)
    • \(\dfrac{x}{2} + 6 - x\)
    • \(5 - 3 \# x\)
    • \(\dfrac{x+3}{2}\)
    • \(-\dfrac{x}{8}\)
  2. Finding Coefficients: What is the coefficient of the term in \(x\) in the following expressions?
    • \(2x - 7\)
    • \(\dfrac{x}{2} - 10\)
    • \(-x + 7\)
    • \(x + 7\)
    • \(-\dfrac{x}{4}\)
  3. Evaluate Expressions: Find the value of \(2x + 7\) for the given values of \(x\): \(x = -1, -4, 0, 5\).
  4. Expression Evaluation: Find the value of \(\dfrac{x}{3} - 9\) for \(x = 0, 3, 9, -12\).
  5. More Evaluation Practice: Find the value of \(\dfrac{-x + 2}{2} + 1\) for \(x = 0, 2, -2, -12\).
  6. Expression with Fractions: Find the value of \(12x - 3\) for \(x = \dfrac{1}{2}, \dfrac{1}{6}, -\dfrac{3}{4}, -\dfrac{1}{4}\).
  7. Decimals in Algebra: Find the value of \(-0.1x + 2.5\) for \(x = 1, -1, 2, -2\).
  8. Compare Expressions: Which of the following expressions has the largest value when \(x = -3\)? (\(x + 10, -x + 10, \dfrac{x}{3} + 10, -\dfrac{x}{3} + 10\))
  9. Commutative Property of Addition: Which of the following shows the commutative property?
  10. Associative Property of Addition: Which of the following shows the associative property?
  11. Simplifying Expressions: Which of the following is equivalent to \(3x + 2 - x + 4\)?
  12. Word Problem: “Two times x plus 20”
  13. Word Problem: “Three times x subtracted from 17”
  14. Cards Problem: Jo has \(x\) cards. Janette has 10 more. Write an expression for Janette’s cards.
  15. Carla’s Toy Cars: Carla had 20 toy cars. She lost \(x\). Write an expression for her remaining cars.
  16. Chris’s Money: Chris had \$200. Spent \$20 on food and bought a book for \(a\) dollars. How much does he have left?
  17. Nathan’s Cards: Nathan had \(x\) cards. Gave half to Tanya, Tanya gave 1/3 of hers to Salma. Write an expression for how many cards Tanya has.

Step-by-Step Algebra Solutions with Explanations

  1. Identifying Algebraic Expressions

    Algebraic expressions contain real numbers, variables, and the four basic operations: \(+, -, \times, \div\).

    Valid examples include: \(2x + 2\), \(\dfrac{x}{2} + 6 - x\), \(\dfrac{x+3}{2}\), \(-\dfrac{x}{8}\).

    Non-examples include expressions with invalid symbols like & or #.

  2. Finding Coefficients

    ExpressionTerm in \(x\)Multiplication FormCoefficient
    \(2x - 7\)\(2x\)\(2 \times x\)2
    \(\dfrac{x}{2} - 10\)\(\dfrac{x}{2}\)\(\dfrac{1}{2} \times x\)\(\dfrac{1}{2}\)
    \(-x + 7\)\(-x\)\(-1 \times x\)-1
    \(x + 7\)\(x\)\(1 \times x\)1
    \(-\dfrac{x}{4}\)\(-\dfrac{x}{4}\)\(-\dfrac{1}{4} \times x\)\(-\dfrac{1}{4}\)

  3. Evaluating \(2x+7\)

    Expression\(x\) valueSubstitutionResult
    \(2x+7\)-1\(2(-1)+7\)5
    \(2x+7\)-4\(2(-4)+7\)-1
    \(2x+7\)0\(2(0)+7\)7
    \(2x+7\)5\(2(5)+7\)17

  4. Evaluating \(\dfrac{x}{3}-9\)

    Expression\(x\) valueSubstitutionResult
    \(\dfrac{x}{3}-9\)0\(0/3-9\)-9
    \(\dfrac{x}{3}-9\)3\(3/3-9\)-8
    \(\dfrac{x}{3}-9\)9\(9/3-9\)-6
    \(\dfrac{x}{3}-9\)-12\(-12/3-9\)-13

  5. Evaluating \(\dfrac{-x+2}{2}+1\)

    Expression\(x\) valueSubstitutionResult
    \(\dfrac{-x+2}{2}+1\)0\((-0+2)/2+1\)2
    \(\dfrac{-x+2}{2}+1\)2\((-2+2)/2+1\)1
    \(\dfrac{-x+2}{2}+1\)-2\((2+2)/2+1\)3
    \(\dfrac{-x+2}{2}+1\)-12\((12+2)/2+1\)8

  6. Evaluating \(12x-3\)

    Expression\(x\) valueSubstitutionResult
    \(12x-3\)\(1/2\)\(12(1/2)-3\)3
    \(12x-3\)\(1/6\)\(12(1/6)-3\)-1
    \(12x-3\)\(-3/4\)\(12(-3/4)-3\)-12
    \(12x-3\)\(-1/4\)\(12(-1/4)-3\)-6

  7. Evaluating \(-0.1x+2.5\)

    Expression\(x\) valueSubstitutionResult
    \(-0.1x+2.5\)1\(-0.1(1)+2.5\)2.4
    \(-0.1x+2.5\)-1\(-0.1(-1)+2.5\)2.6
    \(-0.1x+2.5\)2\(-0.1(2)+2.5\)2.3
    \(-0.1x+2.5\)-2\(-0.1(-2)+2.5\)2.7

  8. Comparing Expressions (at \(x=-3\))

    ExpressionValue of Expression at \(x = - 3\)
    \(x+10\)7
    \(-x+10\)13
    \(\dfrac{x}{3}+10\)9
    \(-\dfrac{x}{3}+10\)11

    The largest value is 13.

  9. Commutative Property: \(a+b=b+a\).
  10. Associative Property: \(a+(b+c)=(a+b)+c\).
  11. Simplifying: \(3x+2-x+4 = 2x+6\).
  12. "Two times x plus 20": \(2x+20\).
  13. "Three times x subtracted from 17": \(17 - 3x\).
  14. Cards: \(x+10\).
  15. Carla: \(20 - x\).
  16. Chris: \(200 - 20 - a\).
  17. Nathan's Cards: Nathan has \(x\). Tanya receives half (\(\frac{1}{2}x\)). She gives away \(\frac{1}{3}\) of her half, leaving her with \(\frac{2}{3}\) of her half: \(\frac{2}{3} \times \frac{1}{2}x = \frac{1}{3}x\).

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