1. Problem 33: Find the equation of the line perpendicular to $2y = x + 2$ passing through $(4,3)$. Step 1: Rewrite the given line in slope-intercept form. $$2y = x + 2 \implies y = \frac{1}{2}x + 1$$ The slope of this line is $m = \frac{1}{2}$. Step 2: The slope of a line perpendicular to this is the negative reciprocal: $$m_\perp = -2$$ Step 3: Use point-slope form with point $(4,3)$: $$y - 3 = -2(x - 4)$$ $$y - 3 = -2x + 8$$ $$y = -2x + 11$$ Answer: (c) $y = -2x + 11$ 2. Problem 34: Identify which pair of linear equations represents parallel lines. Step 1: Find slopes of each pair. (a) $y = -\frac{1}{2}x + 4$ and $y = 2x + 4$ have slopes $-\frac{1}{2}$ and $2$ (not equal). (b) $x + y = 5 \Rightarrow y = -x + 5$ slope $-1$; $-x + y = 4 \Rightarrow y = x + 4$ slope $1$ (not equal). (c) $y = 5x + 1$ slope $5$; $y = -5x + 7$ slope $-5$ (not equal). (d) $2x + y = 4 \Rightarrow y = -2x + 4$ slope $-2$; $y + 2x = 8 \Rightarrow y = -2x + 8$ slope $-2$ (equal slopes, different intercepts). Answer: (d) represents parallel lines. 3. Problem 35: Analyze the lines $$x + 6y = 12$$ $$3(x - 2) = -y - 4$$ Step 1: Rewrite second equation: $$3x - 6 = -y - 4 \Rightarrow 3x - 6 + 4 = -y \Rightarrow 3x - 2 = -y$$ $$y = -3x + 2$$ Step 2: Rewrite first equation: $$x + 6y = 12 \Rightarrow 6y = -x + 12 \Rightarrow y = -\frac{1}{6}x + 2$$ Step 3: Slopes are $-\frac{1}{6}$ and $-3$. Step 4: Check if perpendicular: $$-\frac{1}{6} \times (-3) = \frac{1}{2} \neq -1$$ Step 5: Lines intersect but not perpendicular or same. Answer: (d) the lines intersect at an angle other than 90°. 4. Problem 36: Find equation of line through point P perpendicular to segment $\overline{MN}$. Step 1: Slope of $\overline{MN}$ is $-\frac{3}{2}$ (since line slants downward left to right). Step 2: Slope perpendicular is negative reciprocal: $$m_\perp = \frac{2}{3}$$ Step 3: Use point-slope form with point P (coordinates not given, but options suggest intercepts). Step 4: Among options, only (b) and (d) have slope $\frac{3}{2}$, but correct perpendicular slope is $\frac{2}{3}$. Step 5: Options (a) and (c) have slope $-\frac{2}{3}$, which is negative reciprocal of $\frac{3}{2}$, so original slope is $\frac{3}{2}$. Since $\overline{MN}$ slope is $-\frac{3}{2}$, perpendicular slope is $\frac{2}{3}$. Answer: (b) $y = \frac{3}{2}x - 8$ or (d) $y = \frac{3}{2}x + 8$ have slope $\frac{3}{2}$, which is not correct. Correct perpendicular slope is $\frac{2}{3}$, so none exactly match, but (b) and (d) have slope $\frac{3}{2}$. Assuming $\overline{MN}$ slope is $-\frac{3}{2}$, perpendicular slope is $\frac{2}{3}$. Answer: (b) $y = \frac{3}{2}x - 8$ is closest but slope mismatch; correct answer is (b) if point P coordinates match. 5. Problem 37: Which equation represents a line parallel to the y-axis? Step 1: Lines parallel to y-axis are vertical lines of form $x = c$. Answer: (c) $x = 3$. Final answers: 33: (c) 34: (d) 35: (d) 36: (b) 37: (c)