Equations in One Variable- Grade 6

Grade 6 examples and questions on equations and problems in one variable with detailed solutions, at the bottom of the page, and explanations are presented. If you find that some of the questions are challenging do not skip them, spend time on them and work in groups. We learn math by solving challenging questions.

Algebra Questions and Practice Problems

  1. Which of the following is an equation in one variable?
    1. \(2x + 2\)
    2. \(x + 2 = 4\)
    3. \(8 = x\)
    4. \(4 + \dfrac{x}{3}\)
    5. \(\dfrac{x - 8}{3} = 9\)
    6. \(\dfrac{2x + 7}{3}\)
  2. Which of the following is not an equation?
    1. \(3x - 9\)
    2. \(\dfrac{x}{2} + 4 = 6\)
    3. \(12 - 7 = 5\)
    4. \(8 \times x\)
    5. \(5 = \dfrac{x}{4}\)
  3. Which value of \(x\) satisfies the equation \(2x - 4 = 4\)?
    1. \(x = 0\)
    2. \(x = 4\)
    3. \(x = 2\)
    4. \(x = -2\)
  4. Which value of \(x\) satisfies the equation \(\dfrac{x}{3} - 1 = 2\)?
    1. \(x = -3\)
    2. \(x = 6\)
    3. \(x = -9\)
    4. \(x = 9\)
  5. Solve the following equations:
    1. \(x - 6 = 12\)
    2. \(3 = x + 3\)
    3. \(2 + x = 8\)
    4. \(2x = 16\)
    5. \(\dfrac{x}{3} = 5\)
  6. Which pairs of equations have the same solution?
    1. \(x = 2\) and \(2x = 4\)
    2. \(x + 3 = 6\) and \(x + 4 = 8\)
    3. \(\dfrac{x}{2} = 2\) and \(x = -4\)
    4. \(3x = 9\) and \(x + 1 = 4\)
  7. What value of \(x\) makes the expression \(2x + 6\) equal to 12?
  8. For what value of \(x\) do the expressions \(4x + 6\) and \(2 + 12\) have equal values?
  9. The sum of \(d\) and 23 is 56. What is the value of \(d\)?
  10. Seven subtracted from \(x\) is 41. What is the value of \(x\)?
  11. The product of \(y\) and 6 is 36. What is the value of \(y\)?
  12. The division of \(b\) by 5 is 4. What is the value of \(b\)?
  13. Jacky has \(x\) cards and Jimmy has 23 cards. Together they have 121 cards. How many cards does Jacky have?
  14. Jimmy, Toby, and Dina contributed a total of \$123 to buy a gift. Jimmy contributed \$34 and Dina \$45. How much did Toby contribute?

Solutions to the Above Questions and Problems

  1. Identifying Equations

    An equation in mathematics is a statement that two mathematical expressions are equal. Therefore, an equation must contain an equal sign (=).

    From the given list, only the following are equations:

    1. \(x + 2 = 4\)
    2. \(8 = x\)
    3. \(\dfrac{x - 8}{3} = 9\)

  2. Non-Equations

    According to the definition above, the following are not equations because they lack an equal sign:

    1. \(3x - 9\)
    2. \(8, \; x\)

  3. Checking Solutions

    We test values of \(x\) in the equation \(\;2x - 4 = 4\; \) and compare both sides.

    1. \(x = 0\)
      Left side: \(2(0) - 4 = -4\)
      Right side: \(4\)
      Not equal → not a solution.
    2. \(x = 4\)
      Left side: \(2(4) - 4 = 4\)
      Right side: \(4\)
      Equal → \(x = 4\) is a solution.

  4. Another Equation Test

    Check which values satisfy \(\dfrac{x}{3} - 1 = 2\).

    1. \(x = -3\) → LHS = \(-2\), RHS = \(2\) → not a solution.
    2. \(x = 6\) → LHS = \(1\), RHS = \(2\) → not a solution.
    3. \(x = -9\) → LHS = \(-4\), RHS = \(2\) → not a solution.
    4. \(x = 9\) → LHS = \(2\), RHS = \(2\) → solution.

  5. Solving Basic Linear Equations

    1. Solve \(x - 6 = 12\) → Add 6: \(x = 18\)
    2. Solve \(3 = x + 3\) → Subtract 3: \(x = 0\)
    3. Solve \(2 + x = 8\) → Subtract 2: \(x = 6\)
    4. Solve \(2x = 16\) → Divide by 2: \(x = 8\)
    5. Solve \(\dfrac{x}{3} = 5\) → Multiply by 3: \(x = 15\)

  6. Comparing Equations

    We solve pairs of equations and compare solutions.

    1. \(x = 2\) and \(2x = 4\) → Both give \(x = 2\) → same solutions.
    2. \(x + 3 = 6\) and \(x + 4 = 8\) → Solutions: \(x = 3\) and \(x = 4\) → different.
    3. \(\dfrac{x}{2} = 2\) and \(x = -4\) → Solutions: \(x = 4\) and \(x = -4\) → different.
    4. \(3x = 9\) and \(x + 1 = 4\) → Both give \(x = 3\) → same solutions.

    7. Solving Word Problems with Equations

  7. Find \(x\) if \(2x + 6 = 12\).

    Simplify: \(2x = 6 \; \Rightarrow \; x = 3\). Check: \(2(3) + 6 = 12\) ✔

  8. Solve \(4x + 6 = 2 + 12\).

    Simplify: \(4x = 8 \; \Rightarrow \; x = 2\). Check: \(4(2) + 6 = 14\) ✔

  9. “The sum of \(d\) and 23 is 56.”
    Equation: \(d + 23 = 56\).
    Solution: \(d = 33\).
  10. “Seven subtracted from \(x\) is 41.”
    Equation: \(x - 7 = 41\).
    Solution: \(x = 48\).
  11. “The product of \(y\) and 6 is 36.”
    Equation: \(6y = 36\).
    Solution: \(y = 6\).
  12. “The division of \(b\) by 5 is 4.”
    Equation: \(\dfrac{b}{5} = 4\).
    Solution: \(b = 20\).
  13. Jacky has \(x\) cards, Jimmy has 23, total = 121.
    Equation: \(x + 23 = 121\).
    Solution: \(x = 98\).
  14. Jimmy ($34), Dina ($45), and Toby contribute to a $123 gift.
    Equation: \(34 + 45 + c = 123\).
    Solution: \(c = 44\).

Links and References