1. **Problem 38:** Find which equation is equivalent to $y - 34 = x(x - 12)$.\nExpand the right side: $$y - 34 = x^2 - 12x \Rightarrow y = x^2 - 12x + 34.$$\nNow test each option:\n(a) $$y = (x-17)(x+2) = x^2 + 2x - 17x - 34 = x^2 - 15x - 34$$ (not equivalent)\n(b) $$y = (x - 6)^2 + 2 = x^2 - 12x + 36 + 2 = x^2 - 12x + 38$$ (not equivalent)\n(c) $$y = (x - 6)^2 - 2 = x^2 - 12x + 36 - 2 = x^2 - 12x + 34$$ (equivalent)\n(d) $$y = (x - 17)(x - 2) = x^2 - 2x - 17x + 34 = x^2 - 19x + 34$$ (not equivalent)\n**Answer:** (c)\n\n2. **Problem 39:** Identify which pair of equations cannot solve the system $$\{4x + 2y = 22; -2x + 2y = -8\}.$$\nCheck each pair for equivalence to the original system or contradictions:\n(a) $$\{4x + 2y = 22; 2x - 2y = 8\}$$ Multiply second by -1: $$-2x + 2y = -8$$ original second equation, but here it's $$2x - 2y = 8$$, which changes signs, not identical. But since the left sides differ from the original, this pair is not equivalent.\n(b) $$\{4x + 2y = 22; -4x + 4y = -16\}$$ Multiply original second by 2: $$-4x + 4y = -16$$ matches (b). So (b) is consistent.\n(c) $$\{12x + 6y = 66; 6x - 6y = 24\}$$ Dividing first by 3: $$4x + 2y = 22$$ original first equation. Second differs in sign and coefficients, not matching original second.\n(d) $$\{8x + 4y = 44; -8x + 8y = -8\}$$ Dividing by 2: $$4x + 2y = 22$$ and dividing second by 4: $$-2x + 2y = -8$$ are original equations. So (d) is consistent.\nPair that cannot solve original system is (a) because second equation is not equivalent.\n**Answer:** (a)\n\n3. **Problem 40:** Translate "sixty more than 9 times a number is 375."\nLet number = $h$. Expression is $$9h + 60 = 375.$$\n**Answer:** (a)\n\n4. **Problem 41:** Solve $$\frac{3}{5}(x + 2) = x - 4.$$\nMultiply both sides by 5: $$3(x + 2) = 5(x - 4)$$\n$$3x + 6 = 5x - 20$$\nBring variables to one side: $$6 + 20 = 5x - 3x \Rightarrow 26 = 2x$$\n$$x = 13$$\n**Answer:** (b)\n\n5. **Problem 42:** Solve $$x^2 - 6x = 0$$\nFactor: $$x(x - 6) = 0\Rightarrow x=0 \text{ or } x=6$$\n**Answer:** (c)\n\n6. **Problem 43:** Ages are consecutive even integers: Let youngest = $x$\nOldest = $x + 4$ (since consecutive even integers, difference by 2 each)\nGiven: $$3x = (x + 4) + 48$$\n$$3x = x + 52\Rightarrow 3x - x = 52 \Rightarrow 2x=52 \Rightarrow x=26$$\n**Answer:** (d)\n\n7. **Problem 44:** Sum 47 and difference 15. Let larger number = $L$, smaller = $S$.\nGiven: $$L + S = 47$$ $$L - S = 15$$\nAdd equations: $$2L = 62 \Rightarrow L=31$$\n**Answer:** (c)\n\n8. **Problem 45:** Expression $$\frac{5n}{2n-1}$$ undefined when denominator = 0\nSolve $$2n - 1 = 0 \Rightarrow n = \frac{1}{2}$$\n**Answer:** (d)\n\n9. **Problem 46:** Let Chartered = $x$ students.\nProfessional = $x + 60$\nGraduate = $2x - 50$\nFundamental = $3x$\nSum: $$x + (x+60) + (2x - 50) + 3x = 1424$$\nCombine: $$x + x + 60 + 2x - 50 + 3x = 1424$$\n$$7x + 10 = 1424$$\n$$7x = 1414$$\n$$x = 202$$\n**Answer:** (b)\n\n10. **Problem 47:** Find $x$ such that $$f(x) = h(x)$$ where $$f(x) = \frac{1}{2} x + 3, h(x) = |x|.$$\nConsider two cases: \nCase 1: $x \geq 0$:\n$$\frac{1}{2}x + 3 = x$$\n$$3 = x - \frac{1}{2}x = \frac{1}{2}x$$\n$$x=6$$ (not in the answer choices)\nCase 2: $x < 0$:\n$$\frac{1}{2}x + 3 = -x$$\n$$\frac{1}{2}x + x = -3$$\n$$\frac{3}{2}x = -3$$\n$$x = -2$$\n**Answer:** (a)