Pair of Linear Equations in Two Variables

4

Step 1: Multiply first equation by 3 and second by 4 to eliminate y: 9x + 12y = 75 and 8x - 12y = -12. Step 2: Add the equations: (9x + 12y) + (8x - 12y) = 75 + (-12), giving 17x = 63. Step 3: Solve for x: x = 63/17 = 3.7 (wait, let me recalculate). Actually, let's try eliminating x instead. Step 4: Multiply first by 2 and second by 3: 6x + 8y = 50 and 6x - 9y = -9. Step 5: Subtract: 17y = 59, so y = 59/17. Let me solve this correctly: From 3x + 4y = 25 and 2x - 3y = -3, multiply first by 2, second by 3: 6x + 8y = 50, 6x - 9y = -9. Subtract: 17y = 59, so y ≈ 3.47. Actually checking answer choi

12 years

Step 1: Let son's present age = x years, man's present age = y years. Given: y = 3x. Step 2: After 12 years: man's age = y + 12, son's age = x + 12. Condition: y + 12 = 2(x + 12). Step 3: Substitute y = 3x: 3x + 12 = 2(x + 12) = 2x + 24. Step 4: Simplify: 3x + 12 = 2x + 24, so x = 12. Step 5: Son's current age is 12 years, man's current age is 36 years. Verification: 36 = 3(12) ✓ and after 12 years: 48 = 2(24) ✓

No method works - inconsistent system

Step 1: Check the ratio of coefficients: a₁/a₂ = 1/2, b₁/b₂ = 1/2, but c₁/c₂ = 10/15 = 2/3. Step 2: Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the system is inconsistent. Step 3: This means the lines are parallel and never intersect. Step 4: The second equation 2x + 2y = 15 is equivalent to x + y = 7.5, which contradicts x + y = 10. Step 5: Therefore, no solution exists, and no method can solve this inconsistent system.

4 cm

Step 1: Let width = w cm, length = l cm. Given: l = w + 2. Step 2: Perimeter = 2(l + w) = 20, so l + w = 10. Step 3: Substitute l = w + 2 into l + w = 10: (w + 2) + w = 10. Step 4: Simplify: 2w + 2 = 10, so 2w = 8, giving w = 4. Step 5: Therefore, width = 4 cm and length = 6 cm. Check: Perimeter = 2(4 + 6) = 20 ✓