1. Problem 38: Find which equation is equivalent to $y - 34 = x(x - 12)$. Expand the right side: $$y - 34 = x^2 - 12x$$ Add 34 to both sides: $$y = x^2 - 12x + 34$$ Now let's examine the given choices by expanding them: (a) $y = (x - 17)(x + 2) = x^2 + 2x - 17x - 34 = x^2 - 15x - 34$ (b) $y = (x - 6)^2 + 2 = (x^2 - 12x + 36) + 2 = x^2 - 12x + 38$ (c) $y = (x - 6)^2 - 2 = (x^2 - 12x + 36) - 2 = x^2 - 12x + 34$ (d) $y = (x - 17)(x - 2) = x^2 - 2x - 17x + 34 = x^2 - 19x + 34$ Comparing with $y = x^2 - 12x + 34$, only option (c) matches. 2. Problem 39: Identify which pair of equations could not be used to solve the system $$\begin{cases} 4x + 2y = 22 \\ -2x + 2y = -8 \end{cases}$$ Check each choice to see if it represents the original system or an equivalent system: (a) System: $$\begin{cases} 4x + 2y = 22 \\ 2x - 2y = 8 \end{cases}$$ Second equation differs in sign from original second equation; this system is not equivalent. (b) System: $$\begin{cases} 4x + 2y = 22 \\ -4x + 4y = -16 \end{cases}$$ Second equation is the original second equation multiplied by 2. Therefore equivalent. (c) System: $$\begin{cases} 12x + 6y = 66 \\ 6x - 6y = 24 \end{cases}$$ Both equations are original equations multiplied by 3 and -3 respectively. So equivalent. (d) System: $$\begin{cases} 8x + 4y = 44 \\ -8x + 8y = -8 \end{cases}$$ Both equations are original multiplied by 2 and -4 respectively. So equivalent. Answer: (a) could not be used. 3. Problem 40: Translate "sixty more than 9 times a number is 375". "9 times a number" is $9h$ "sixty more than" means adding 60: $9h + 60 = 375$ Hence correct equation: (a) 4. Problem 41: Solve $$\frac{3}{5}(x + 2) = x - 4$$ Multiply both sides by 5: $$3(x + 2) = 5x - 20$$ Expand: $$3x + 6 = 5x - 20$$ Bring terms to one side: $$6 + 20 = 5x - 3x$$ $$26 = 2x$$ $$x = 13$$ Answer: (b) 5. Problem 42: Solve $$x^2 - 6x = 0$$ Factor: $$x(x - 6) = 0$$ Solutions: $$x=0$$ or $$x=6$$ Answer: (c) 6. Problem 43: Ages are consecutive even integers. Let youngest be $x$. Oldest is $x+4$ (since even integers spaced by 2; three consecutive even integers are $x$, $x+2$, $x+4$). Given: $3x = (x + 4) + 48$ Simplify: $$3x = x + 52$$ $$3x - x = 52$$ $$2x = 52$$ $$x = 26$$ Answer: (d) 7. Problem 44: Sum of two numbers is 47 and difference is 15. Let the numbers be $x$ (larger) and $y$ (smaller). $$x + y = 47$$ $$x - y = 15$$ Add both: $$2x = 62$$ $$x = 31$$ Answer: (c) 8. Problem 45: Expression undefined when denominator zero. $$2n - 1 = 0$$ $$2n = 1$$ $$n = \frac{1}{2}$$ Answer: (d) 9. Problem 46: Let Chartered level class have $C$ students. Professional: $C + 60$ Graduate: $2C - 50$ Fundamental: $3C$ Total students: $$C + (C+60) + (2C - 50) + 3C = 1424$$ Simplify: $$C + C + 60 + 2C - 50 + 3C = 1424$$ $$7C + 10 = 1424$$ $$7C = 1414$$ $$C = 202$$ Answer: (b) 10. Problem 47: Solve $f(x) = h(x)$ where $$f(x) = \frac{1}{2}x + 3$$ $$h(x) = |x|$$ Case 1: $x \geq 0$ $$\frac{1}{2}x + 3 = x$$ $$3 = x - \frac{1}{2}x = \frac{1}{2}x$$ $$x = 6$$ Since $x \geq 0$, $x=6$ valid. Case 2: $x < 0$ $$\frac{1}{2}x + 3 = -x$$ $$\frac{1}{2}x + x = -3$$ $$\frac{3}{2}x = -3$$ $$x = -2$$ Check $x < 0$, $-2$ valid. Given options include -2 and 2 only, with solutions $x = -2$ and $x=6$ (not 2). Correct answer: (a) $-2$ Slug: "equation equivalences" Subject: "algebra" Desmos: {"latex":"y=x(x-12)","features":{"intercepts":true,"extrema":true}} q_count:10