This is a two-step linear equation. To solve 3x + 6 = 15, first subtract 6 from both sides to get 3x = 9. Then divide both sides by 3 to isolate x: x = 3. The answer is 3.

The correct answer is C (8). Solve the linear equation: 4y − 7 = 2y + 9. Subtract 2y from both sides: 2y − 7 = 9. Add 7 to both sides: 2y = 16. Divide by 2: y = 8. D (16) is the most common error — students correctly reach 2y = 16 but forget to divide by 2, reporting 16 as the answer. B (2) results from dividing 16 by 8 (the answer) instead of by 2 — a circular error. A (1) comes from a more serious arithmetic mistake earlier in the process. Always check your answer by substituting back: 4(8) − 7 = 25 = 2(8) + 9 ✓.

This is a linear equations question testing one-variable solving. Choice A (−9) is correct — subtract 4x from both sides: −x − 7 = 2. Add 7: −x = 9. Divide both sides by −1: x = −9. Choice B (−5) results from a subtraction error when combining terms, arriving at −x = 5 instead of −x = 9. Choice C (5) solves −x = −5 (a sign error earlier in the process), giving x = 5. Choice D (9) correctly arrives at −x = 9 but then forgets to divide by −1, reporting x = 9 instead of x = −9. Pro tip: When you reach the step −x = [number], always divide both sides by −1 and flip the sign. The equation is NOT solved until x (not −x) is isolated.

This is a literal equations question testing multi-step algebraic isolation. Choice C (√(2K/m)) is correct — isolate v: K = (1/2)mv² → 2K = mv² → v² = 2K/m → v = √(2K/m). Choice A (2K/m) correctly isolates v² but forgets to take the square root — stopping one step early. Choice B (√(K/2m)) divides K by 2m instead of multiplying: student moves the (1/2) to the denominator rather than multiplying both sides by 2, giving v² = K/(m/2)... actually that gives K/(m/2) = 2K/m, which is correct. B may come from: K = (1/2)mv² → K/m = (1/2)v² → v² = K/(m/2) → student writes √(K/2m) incorrectly. Most likely B = student writes v = √(K/2m) from not doubling K. Choice D (√K / 2m) takes the square root of K alone without dividing by m. Pro tip: When isolating a squared variable, handle all multiplication/division first, then take the square root last. Write each step: 2K = mv² (multiply both sides by 2), then v² = 2K/m (divide both sides by m), then v = √(2K/m) (square root both sides).

This is a linear equation with variable terms on both sides. To solve 2x - 3 = x + 7, subtract x from both sides: x - 3 = 7. Add 3 to both sides: x = 10. The solution is x = 10.

This is a two-step linear equation. To solve 2x + 4 = 12, first subtract 4 from both sides to get 2x = 8. Then divide both sides by 2 to isolate x: x = 4. The answer is 4.

This is a basic linear equation x - 9 = 4 representing a savings account change, with x as the balance or related value. Add 9 to both sides to isolate x: x = 4 + 9. This simplifies to x = 13. Therefore, the value of x is 13, which is choice C. Choice D (5) could be from subtracting 9 instead of adding, getting 4 - 9 = -5, but that's choice B; careful sign handling is key.

This is a linear equation with distribution: 3(x + 4) = 27 marks up an item with a fee, where x is the base value. Distribute the 3: 3x + 12 = 27. Subtract 12 from both sides: 3x = 15. Divide by 3: x = 5. Therefore, the value of x is 5, which is choice B. Choice C (9) might result from dividing 27 by 3 without distributing, getting x + 4 = 9 and forgetting to subtract 4.

This equation requires distributing first. To solve 2(x + 7) = 30, distribute the 2: 2x + 14 = 30. Subtract 14 from both sides: 2x = 16. Divide by 2: x = 8. Choice B (x = 23) likely forgot to distribute the 2.

This equation has variables on both sides representing equal savings. To solve 3x + 2 = x + 14, subtract x from both sides: 2x + 2 = 14. Subtract 2: 2x = 12. Divide by 2: x = 6.