1. Problem: Find which equation is equivalent to $y - 34 = x(x - 12)$.\nStep 1: Expand the right side: $$y - 34 = x^2 - 12x$$\nStep 2: Add 34 to both sides: $$y = x^2 - 12x + 34$$\nStep 3: Try to rewrite $y = x^2 - 12x + 34$ in forms given:\n(a) Expand $(x - 17)(x + 2)=x^2 + 2x - 17x - 34 = x^2 - 15x - 34$ (No)\n(b) Expand $(x - 6)^2 + 2 = (x^2 - 12x + 36) + 2 = x^2 -12x + 38$ (No)\n(c) Expand $(x - 6)^2 - 2 = (x^2 -12x + 36) - 2 = x^2 - 12x + 34$ (Yes)\n(d) Expand $(x - 17)(x - 2) = x^2 - 19x + 34$ (No)\nAnswer: (c)\n\n2. Problem: Which pair of equations could NOT be used for solving the system\n$\{4x + 2y = 22, \ -2x + 2y = -8\}$?\nStep 1: Understand that pairs must be equivalent or multiples that preserve solution.\nCheck each pair for equivalence:\n(a) Second equation changed sign of first? Actually, $2x - 2y = 8$ is not equivalent to $-2x + 2y = -8$ (sign changes everywhere), so (a) no.\n(b) Multiply second original equation by 2: $-4x + 4y = -16$ matches (b) (Yes)\n(c) Multiply original equations by 3: first $12x + 6y = 66$, second $-6x + -6y = -24$ but in (c) second is $6x - 6y = 24$, sign changed so not equivalent. So (c) no.\n(d) Multiply first equation by 2: $8x + 4y = 44$, multiply second by -4: $8x - 8y = 32$, but (d) has $-8x + 8y = -8$, not matching, so no.\nAnswer: (a) could not be used.\n\n3. Problem: Translate 'sixty more than 9 times a number is 375'.\nStep 1: Let the number be $h$.\nStep 2: "Nine times a number" is $9h$.\nStep 3: "Sixty more than" means add 60: $9h + 60$.\nStep 4: Set equal to 375: $9h + 60 = 375$.\nAnswer: (a)\n\n4. Problem: Solve $\frac{3}{5}(x + 2) = x - 4$.\nStep 1: Multiply both sides by 5: $3(x + 2) = 5x - 20$.\nStep 2: Expand left: $3x + 6 = 5x - 20$.\nStep 3: Move $3x$ to right: $6 = 5x - 20 - 3x = 2x - 20$.\nStep 4: Add 20: $26 = 2x$.\nStep 5: Divide by 2: $x = 13$.\nAnswer: (b)\n\n5. Problem: Solve $x^2 - 6x = 0$.\nStep 1: Factor: $x(x - 6) =0$.\nStep 2: Set each factor zero: $x=0$ or $x - 6=0 \Rightarrow x=6$.\nAnswer: (c)\n\n6. Problem: Three brothers' ages are consecutive even integers. Three times youngest exceeds oldest by 48. Find youngest.\nStep 1: Let youngest be $x$, next $x + 2$, oldest $x + 4$.\nStep 2: Expression: $3x - (x + 4) = 48$.\nStep 3: Simplify: $3x - x - 4 = 48 \Rightarrow 2x - 4 = 48$.\nStep 4: Add 4: $2x = 52$.\nStep 5: Divide by 2: $x = 26$.\nAnswer: (d)\n\n7. Problem: Sum of two numbers is 47 and difference is 15. Find larger number.\nStep 1: Let numbers be $x$ and $y$, with $x > y$.\nStep 2: Equations: $x + y = 47$, $x - y = 15$.\nStep 3: Add equations: $2x = 62 \Rightarrow x = 31$.\nAnswer: (c)\n\n8. Problem: Find $n$ that makes $\frac{5n}{2n - 1}$ undefined.\nStep 1: Denominator zero: $2n - 1 = 0$.\nStep 2: Solve: $2n = 1 \Rightarrow n = \frac{1}{2}$.\nAnswer: (d)\n\n9. Problem: ICMA classes total 1424 students. Given relations: Chartered = $c$, Professional = $c + 60$, Graduate = $2c - 50$, Fundamental = $3c$. Find $c$.\nStep 1: Write total: $c + (c + 60) + (2c - 50) + 3c = 1424$.\nStep 2: Simplify: $c + c + 60 + 2c - 50 + 3c = 1424$, sum $c$s: $7c + 10 = 1424$.\nStep 3: Subtract 10: $7c = 1414$.\nStep 4: Divide by 7: $c = 202$.\nAnswer: (b)\n\n10. Problem: For $f(x) = \frac{1}{2}x + 3$ and $h(x) = |x|$, find $x$ with $f(x) = h(x)$.\nStep 1: Solve $\frac{1}{2}x + 3 = |x|$.\nStep 2: Consider two cases:\nCase 1: $x \geq 0$, then $|x| = x$. Equation: $\frac{1}{2}x + 3 = x$.\nSubtract $\frac{1}{2}x$: $3 = \frac{1}{2}x$, so $x=6$. Not in options.\nCase 2: $x < 0$, then $|x| = -x$. Equation: $\frac{1}{2}x + 3 = -x$.\nAdd $x$: $\frac{3}{2}x + 3 = 0$.\nSubtract 3: $\frac{3}{2}x = -3$.\nMultiply both sides by $\frac{2}{3}$: $x = -2$.\nAnswer: (a)\n