1. Problem 41: Find the rate of change for the resale value of a refrigerator given the data points from (0, 600) to (15, 0). Step 1: Identify two points on the line: $(0,600)$ and $(15,0)$. Step 2: Use the slope formula $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 600}{15 - 0} = \frac{-600}{15} = -40.$$ Step 3: The rate of change is $-40$ dollars per year. Step 4: To check the options, calculate the drop over 6 years: $$-40 \times 6 = -240.$$ None of the options exactly match $-40$ per year or $-240$ per 6 years, but the closest is option (a) which states $-45$ per year and $270$ every 6 years. 2. Problem 42: Find the equation of line A passing through $(5,3)$ and parallel to line B: $5x + y = 6$. Step 1: Rewrite line B in slope-intercept form: $$y = -5x + 6.$$ The slope of line B is $-5$. Step 2: Since line A is parallel to line B, it has the same slope $m = -5$. Step 3: Use point-slope form for line A: $$y - 3 = -5(x - 5).$$ Step 4: Simplify: $$y - 3 = -5x + 25 \Rightarrow y = -5x + 28.$$ Step 5: The equation of line A is $y = -5x + 28$, which matches option (b). 3. Problem 43: Determine the true statement about lines A, B, and C given: $$A: x + 2y = 6,$$ $$B: 6x + 3y = 12,$$ $$C: -x + 2y = 10.$$ Step 1: Find slopes of each line. - For A: $$2y = -x + 6 \Rightarrow y = -\frac{1}{2}x + 3,$$ slope $m_A = -\frac{1}{2}$. - For B: $$3y = -6x + 12 \Rightarrow y = -2x + 4,$$ slope $m_B = -2$. - For C: $$2y = x + 10 \Rightarrow y = \frac{1}{2}x + 5,$$ slope $m_C = \frac{1}{2}$. Step 2: Check relationships: - Are B and C perpendicular? $m_B \times m_C = -2 \times \frac{1}{2} = -1$, so yes, B is perpendicular to C. - Are C and A perpendicular? $m_C \times m_A = \frac{1}{2} \times -\frac{1}{2} = -\frac{1}{4} \neq -1$. - Are A and B parallel? $m_A = -\frac{1}{2}$ and $m_B = -2$ are not equal. - Are C and A parallel? $\frac{1}{2} \neq -\frac{1}{2}$. Step 3: The true statement is (a) $B \perp C$. 4. Problem 44: Determine the relationship between lines given by equations: $$3y + 6 = 2x$$ and $$2y - 3x = 6.$$ Step 1: Rewrite both in slope-intercept form. - First line: $$3y = 2x - 6 \Rightarrow y = \frac{2}{3}x - 2,$$ slope $m_1 = \frac{2}{3}$. - Second line: $$2y = 3x + 6 \Rightarrow y = \frac{3}{2}x + 3,$$ slope $m_2 = \frac{3}{2}$. Step 2: Check if slopes are equal (parallel) or negative reciprocals (perpendicular). - $m_1 = \frac{2}{3}$, $m_2 = \frac{3}{2}$. - $m_1 \times m_2 = \frac{2}{3} \times \frac{3}{2} = 1 \neq -1$, so not perpendicular. - Slopes are not equal, so not parallel. Step 3: The lines intersect but are not perpendicular, so answer is (d). Final answers: 41: Rate of change is $-40$ per year (closest option (a)). 42: Equation of line A is $y = -5x + 28$ (option (b)). 43: $B \perp C$ (option (a)). 44: Lines intersect but are not perpendicular (option (d)).