Solve Linear Equations with Parentheses & Practice Questions

Comprehensive tutorial and practice questions on solving linear equations with parentheses using the distributive property. Detailed solutions, matched exercises with answers, and multiple-choice problems are included.

Solving Equations Using the Distributive Law

This section presents detailed examples and exercises on solving linear equations with parentheses using the distributive property. Solutions include step-by-step reasoning with explanatory notes.

To remove parentheses, apply the distributive law:

\[ a(b + c) = ab + ac \]

Examples with Solutions

Example 1: Solve \( -2(x + 3) = x + 6 \)
  • Given: \( -2(x+3)=x+6 \)
  • Use the distributive law: \[ -2x - 6 = x + 6 \]
  • Add 6 to both sides: \[ -2x - 6 + 6 = x + 6 + 6 \implies -2x = x + 12 \]
  • Subtract \(x\) from both sides: \[ -2x - x = 12 \implies -3x = 12 \]
  • Divide both sides by \(-3\): \[ x = -4 \]
  • Check:
    Left side: \( -2(-4+3) = 2 \)
    Right side: \( -4 + 6 = 2 \)
  • Conclusion: \( x = -4 \)

Matched Exercise 1: Solve \( -3(-x+3)=x-7 \)

Example 2: Solve \( -3(-x - 6) = 3x - 23 \)
  • Given: \[ -3(-x - 6)=3x - 23 \]
  • Apply the distributive law: \[ 3x + 18 = 3x - 23 \]
  • Subtract 18 from both sides: \[ 3x = 3x - 41 \]
  • Subtract \(3x\) from both sides: \[ 0 = -41 \]
  • Conclusion: No real number satisfies the equation. The equation has no solution.

Matched Exercise 2: Solve \( 4(-x + 3) = -4x - 7 \)

Example 3: Solve \( -7(x - 6) - 3x - 3 = 3(x + 5) - 2x \)
  • Expand all parentheses: \[ -7x + 42 - 3x - 3 = 3x + 15 - 2x \]
  • Group like terms: \[ -10x + 39 = x + 15 \]
  • Subtract 39 from both sides: \[ -10x = x - 24 \]
  • Subtract \(x\) from both sides: \[ -11x = -24 \]
  • Divide by \(-11\): \[ x = \frac{24}{11} \]
  • Check: Both sides evaluate to \( \frac{189}{11} \).
  • Conclusion: \( x = \frac{24}{11} \)

Matched Exercise 3: Solve \( -5(x - 4) - x + 23 = 5(x - 5) - x \)

Example 4: Solve \( -\,\frac{2(x - 6)}{7} - \frac{x - 3}{2} = -x \)
  • Multiply both sides by the LCD \(14\): \[ 14\left[-\frac{2(x - 6)}{7} - \frac{x - 3}{2}\right] = 14(-x) \]
  • Simplify: \[ -4(x-6) - 7(x-3) = -14x \]
  • Expand and combine terms: \[ -11x + 45 = -14x \]
  • Add \(14x\) and subtract 45: \[ 3x = -45 \]
  • Divide by 3: \[ x = -15 \]
  • Check: Both sides equal 15.
  • Conclusion: \( x = -15 \)

Matched Exercise 4: Solve \( -3\,\frac{(x + 4)}{4} - x - 2 = \frac{x - 4}{3} - x \)

Answers to Matched Exercises

  • Matched Exercise 1 Answer: \( x = 1 \)
  • Matched Exercise 2 Answer: No solution (Identity contradiction: \( 12 = -7 \))
  • Matched Exercise 3 Answer: \( x = \frac{34}{5} \) (or \( 6.8 \))
  • Matched Exercise 4 Answer: \( x = -\frac{44}{13} \)

Multiple-Choice Practice Questions

Select the correct answer for each question.

Question 1

Solve the equation:

\[-2x + 6 = 4x - 2\]

  • a: no solutions
  • b: 0
  • c: 3
  • d: \( \tfrac{1}{2} \)
  • e: \( \tfrac{4}{3} \)

Question 2

Solve the equation:

\[-(-x - 5) = x - 2\]

  • a: no solutions
  • b: -5
  • c: 3
  • d: \( \tfrac{1}{2} \)
  • e: \( \tfrac{4}{3} \)

Question 3

What is the solution to the equation:

\[-(-x - 5) + 5(x - 9) = 2(x + 8) - (2x + 5)\]

  • a: 5
  • b: no solutions
  • c: \( \tfrac{17}{2} \)
  • d: \( \tfrac{15}{2} \)
  • e: -8

Question 4

Solve the equation:

\[-\dfrac{3x - 1}{3} - \dfrac{2(x - 8)}{5} = \dfrac{x + 8}{5} - 3\]

  • a: -10
  • b: 15
  • c: \( \tfrac{17}{2} \)
  • d: \( \tfrac{37}{12} \)
  • e: no solutions

Question 5

What is the solution to the equation:

\[\dfrac{2}{7} - \dfrac{x + 8}{4} = \dfrac{3(x - 3)}{5} - \dfrac{3}{2}\]

  • a: no solutions
  • b: 0
  • c: \( \tfrac{17}{111} \)
  • d: \( \tfrac{37}{119} \)
  • e: \( \tfrac{222}{119} \)

Answers to Multiple-Choice Questions
  • Question 1: e
  • Question 2: a
  • Question 3: c
  • Question 4: d
  • Question 5: e