Question

a. 4x - 3y = 6
b. 4x + 3y = 6
c. 4x - 3y = -6
d. 4x + 3y = -6

Explanation:

Step1: Find two points on the line

From the graph, we can see that the line passes through \((-1.5, -4)\) (or \((-\frac{3}{2}, -4)\)) and \((0, -2)\). Let's use two clear points, say when \(x = 0\), \(y=-2\) and when \(y = 0\), from the graph, when \(y = 0\), \(x=\frac{3}{2}\) (or \(1.5\)). Wait, actually, let's calculate the slope first. The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((0, -2)\) and \((\frac{3}{2}, 0)\). Then \(m=\frac{0 - (-2)}{\frac{3}{2}-0}=\frac{2}{\frac{3}{2}}=\frac{4}{3}\).

Step2: Use slope - intercept form \(y=mx + b\)

We know \(b=-2\) (the y - intercept). So the equation is \(y=\frac{4}{3}x-2\). Multiply both sides by 3 to get \(3y = 4x-6\), then rearrange to \(4x-3y = 6\)? Wait, no, wait. Wait, let's check the points. Wait, maybe I made a mistake in the points. Let's take another approach. Let's plug in the point \((0, -2)\) into each equation.

For option A: \(4(0)-3(-2)=0 + 6=6\), which is \(6 = 6\), so \((0, -2)\) satisfies A.

For option B: \(4(0)+3(-2)=-6
eq6\), so B is out.

For option C: \(4(0)-3(-2)=6
eq - 6\), so C is out.

For option D: \(4(0)+3(-2)=-6
eq - 6\)? Wait, \(4(0)+3(-2)=-6\), so \((0, -2)\) satisfies D? Wait, no, let's check another point. Let's take the x - intercept. From the graph, the line crosses the x - axis at \(x =-\frac{3}{2}\) (or \(-1.5\))? Wait, no, wait the graph: when \(y = 0\), where does it cross? Wait, looking at the graph, the line goes from the bottom left to the top right, so it has a positive slope. Let's take two points: when \(x=-3\), \(y=-6\)? Wait, no, let's use the equation of option A: \(4x-3y = 6\). Let's solve for \(y\): \(y=\frac{4x - 6}{3}=\frac{4}{3}x-2\). When \(x = 0\), \(y=-2\) (correct, as per the graph). When \(y = 0\), \(4x=6\), \(x=\frac{3}{2}\) (1.5), which matches the graph (the line crosses the x - axis at \(x = 1.5\)). Now check option D: \(4x + 3y=-6\), \(y=\frac{-4x - 6}{3}=-\frac{4}{3}x - 2\), which has a negative slope, but the line in the graph has a positive slope, so D is out. Option B: \(4x+3y = 6\), \(y=\frac{-4x + 6}{3}=-\frac{4}{3}x + 2\), negative slope, out. Option C: \(4x-3y=-6\), \(y=\frac{4x + 6}{3}=\frac{4}{3}x + 2\), y - intercept at \(y = 2\), but our graph has y - intercept at \(y=-2\), so out. Option A: \(4x-3y = 6\), \(y=\frac{4x - 6}{3}=\frac{4}{3}x-2\), y - intercept at \(y=-2\), positive slope, which matches the graph.

Answer:

A. \(4x - 3y = 6\)