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Quiz: Solving Linear Equations

Test your understanding of solving linear equations in one variable with these questions.

1. What is the first step in solving the equation 3x + 7 = 22?

  1. Divide both sides by 3
  2. Multiply both sides by 3
  3. Subtract 7 from both sides
  4. Add 7 to both sides
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The correct answer is C. When solving a two-step equation like 3x + 7 = 22, the standard strategy is to first eliminate the constant term by subtracting 7 from both sides, giving 3x = 15. Then you can divide by 3 to isolate x. Option A skips the first step and would result in x + 7/3 = 22/3. Options B and D would move in the wrong direction.

Concept Tested: Two-Step Equations

See: Two-Step Equations

2. Which operation must you perform to solve x/4 = -6?

  1. Divide both sides by 4
  2. Add 4 to both sides
  3. Subtract 4 from both sides
  4. Multiply both sides by 4
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The correct answer is D. To solve x/4 = -6, you need to eliminate the division by 4. The inverse operation of division is multiplication, so multiply both sides by 4 to get x = -24. Option A would result in x/16 = -3/2. Options B and C don't address the division operation.

Concept Tested: One-Step Equations

See: One-Step Equations

3. When solving -5x + 3 = 18, what is the value of x?

  1. x = 4
  2. x = -4
  3. x = -3
  4. x = 3
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The correct answer is C. First, subtract 3 from both sides: -5x = 15. Then divide both sides by -5: x = -3. Option A results from forgetting the negative sign on the coefficient. Option B comes from dividing incorrectly. You can verify: -5(-3) + 3 = 15 + 3 = 18 ✓

Concept Tested: Two-Step Equations

See: Two-Step Equations

4. What type of solution does the equation 2(x + 3) = 2x + 6 have?

  1. No solution (contradiction)
  2. One solution: x = 0
  3. One solution: x = 3
  4. Infinitely many solutions (identity)
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The correct answer is D. Distributing gives 2x + 6 = 2x + 6. Subtracting 2x from both sides results in 6 = 6, which is always true. This is an identity equation, meaning every real number is a solution. Option A would result from a false statement like 0 = 5. Options B and C incorrectly assume a single solution exists.

Concept Tested: Identity Equation

See: Identity Equations

5. To solve the equation with fractions x/3 + x/4 = 7, what is the best first step?

  1. Add the fractions on the left side
  2. Multiply both sides by 12 (the LCD)
  3. Divide both sides by 7
  4. Multiply both sides by 3
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The correct answer is B. The most efficient method is to multiply every term by the least common denominator (LCD) of 3 and 4, which is 12. This clears all fractions: 12·(x/3) + 12·(x/4) = 12·7, giving 4x + 3x = 84, so 7x = 84 and x = 12. Option A works but is more complicated. Options C and D don't eliminate the fractions.

Concept Tested: Equations with Fractions

See: Equations with Fractions

6. Solve for w in the perimeter formula P = 2l + 2w. Which is correct?

  1. w = (P - 2l)/2
  2. w = P - 2l
  3. w = 2(P - l)
  4. w = (P + 2l)/2
Show Answer

The correct answer is A. To solve for w, first subtract 2l from both sides: P - 2l = 2w. Then divide by 2: w = (P - 2l)/2. This is formula manipulation, a key skill for literal equations. Option B forgets to divide by 2. Option C incorrectly distributes. Option D uses addition instead of subtraction.

Concept Tested: Literal Equations / Formula Manipulation

See: Literal Equations

7. When solving 5x - 7 = 2x + 8, what should you do after collecting variable terms on one side?

  1. Divide by the coefficient
  2. Subtract the constant from both sides
  3. Add the constant to both sides
  4. Multiply by the reciprocal
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The correct answer is C. After subtracting 2x from both sides, you get 3x - 7 = 8. The next step is to add 7 to both sides to isolate the variable term: 3x = 15. Then you divide by 3 to get x = 5. Option A skips collecting constants. Option B would give 3x = 1. While option D can work, it's the same as dividing and not the typical next step after collecting variables.

Concept Tested: Variables on Both Sides

See: Variables on Both Sides

8. What happens when you solve x + 5 = x - 2?

  1. You get x = -7
  2. You get x = 3
  3. You get 5 = -2, which is false (no solution)
  4. You get 0 = 0, which is true (infinite solutions)
Show Answer

The correct answer is C. Subtracting x from both sides gives 5 = -2, which is a false statement. This means the equation is a contradiction with no solution. The original equation shows a number that is 5 more than x equals a number that is 2 less than x, which is impossible. Option D would indicate an identity. Options A and B incorrectly assume a solution exists.

Concept Tested: Contradiction

See: Contradictions

9. To solve 0.5x + 1.2 = 3.7, what power of 10 should you multiply by to clear decimals?

  1. 1 (no multiplication needed)
  2. 10
  3. 100
  4. 1000
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The correct answer is B. The decimal with the most places is 0.5, 1.2, and 3.7, all with one decimal place. Multiplying by 10 converts these to whole numbers: 10(0.5x) + 10(1.2) = 10(3.7) gives 5x + 12 = 37, then 5x = 25, so x = 5. Option C or D would work but creates unnecessarily large numbers. Option A leaves the decimals.

Concept Tested: Equations with Decimals

See: Equations with Decimals

10. The sum of three consecutive integers is 72. If n represents the first integer, which equation models this situation?

  1. n + 2n + 3n = 72
  2. n + (n + 1) + (n + 2) = 72
  3. n + (n + 2) + (n + 4) = 72
  4. 3n = 72
Show Answer

The correct answer is B. Consecutive integers differ by 1, so if n is the first, then n + 1 is the second, and n + 2 is the third. Their sum is n + (n + 1) + (n + 2) = 72. Solving gives 3n + 3 = 72, so 3n = 69, and n = 23. The integers are 23, 24, and 25. Option A uses multiples instead of consecutive integers. Option C describes consecutive even integers. Option D omits the +1 and +2.

Concept Tested: Applications of Linear Equations

See: Applications of Linear Equations