Question

the graph shows the relationship between the number of shares of stock from company a, x, and the number of shares of stock from company b, y, that simone can purchase. which equation could represent this relationship? a) y = 8x + 12 b) 8x + 12y = 480 c) y = 12x + 8 d) 12x + 8y = 480

Explanation:

Step1: Identify Intercepts

The graph intersects the \(x\)-axis at \((60, 0)\) (when \(y = 0\), \(x = 60\)) and the \(y\)-axis at \((0, 40)\) (when \(x = 0\), \(y = 40\)).

Step2: Check Option D

For option D: \(12x + 8y = 480\).

• When \(x = 0\): \(12(0) + 8y = 480 \implies 8y = 480 \implies y = 60\). Wait, no, wait—wait, earlier intercept was \((0, 40)\). Wait, maybe miscalculation. Wait, let's re - evaluate. Wait, maybe the slope. The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 40}{60 - 0}=-\frac{2}{3}\). Let's rewrite option D in slope - intercept form (\(y=mx + b\)):

\(12x+8y = 480\implies8y=-12x + 480\implies y=-\frac{12}{8}x + 60\implies y =-\frac{3}{2}x+60\). No, that's not matching. Wait, maybe I made a mistake in intercepts. Wait, the \(x\)-axis is Company A (\(x\)) and \(y\)-axis is Company B (\(y\)). Wait, the graph: when \(x = 0\), \(y = 40\); when \(y = 0\), \(x = 60\). Let's check option B: \(8x + 12y=480\).

• When \(x = 0\): \(12y = 480\implies y = 40\) (matches \(y\)-intercept).
• When \(y = 0\): \(8x=480\implies x = 60\) (matches \(x\)-intercept).

Wait, earlier mistake with option D. Let's re - check option B: \(8x + 12y = 480\).
Slope - intercept form: \(12y=-8x + 480\implies y=-\frac{8}{12}x + 40\implies y=-\frac{2}{3}x + 40\). The slope between \((0,40)\) and \((60,0)\) is \(\frac{0 - 40}{60 - 0}=-\frac{2}{3}\), which matches.

Wait, but let's check the options again. Wait, the original options:
A: \(y = 8x+12\) (positive slope, graph has negative slope, eliminate).
C: \(y = 12x + 8\) (positive slope, eliminate).
Now between B (\(8x + 12y = 480\)) and D (\(12x + 8y = 480\)).

For B: when \(x = 60\), \(8(60)+12y = 480\implies480+12y = 480\implies12y = 0\implies y = 0\) (correct). When \(y = 40\), \(8x+12(40)=480\implies8x + 480 = 480\implies8x = 0\implies x = 0\) (correct).

For D: \(12x+8y = 480\). When \(x = 0\), \(8y = 480\implies y = 60\) (not 40, incorrect). When \(y = 0\), \(12x = 480\implies x = 40\) (not 60, incorrect).

Wait, I had a mistake earlier. So the correct option is B? No, wait, let's recalculate the slope. The two points are \((0,40)\) and \((60,0)\). The slope \(m=\frac{0 - 40}{60 - 0}=-\frac{2}{3}\). Let's find the equation of the line using intercept form \(\frac{x}{a}+\frac{y}{b}=1\), where \(a = 60\) (x - intercept) and \(b = 40\) (y - intercept). So \(\frac{x}{60}+\frac{y}{40}=1\). Multiply both sides by 240 (LCM of 60 and 40) to eliminate denominators: \(4x + 6y=240\). Wait, that's not matching. Wait, maybe the intercepts are different. Wait, the grid: each square is, let's see, the \(x\)-axis goes from 0 to 100, with 10 - unit intervals. The \(y\)-axis from 0 to 50, 10 - unit intervals. The line goes from (0,40) to (60,0). Let's check the equation \(8x + 12y = 480\): divide both sides by 8: \(x+\frac{12}{8}y = 60\implies x + 1.5y=60\). When \(x = 60\), \(60+1.5y = 60\implies y = 0\). When \(y = 40\), \(x+1.5\times40=60\implies x + 60 = 60\implies x = 0\). Correct.

Another way: use two - point form. The two points are \((0,40)\) and \((60,0)\). The slope \(m=\frac{0 - 40}{60 - 0}=-\frac{2}{3}\). The equation is \(y-40=-\frac{2}{3}(x - 0)\implies y=-\frac{2}{3}x + 40\). Multiply both sides by 12: \(12y=-8x + 480\implies8x + 12y = 480\), which is option B.

Answer:

B. \(8x + 12y = 480\)