Question

which graph matches the equation $y + 6 = \frac{3}{4}(x + 4)$? graphs are shown, one with points (4,0), (0,-3), (-4,-6) and another with (6,5)

Explanation:

Step1: Identify the form of the equation

The given equation \( y + 6=\frac{3}{4}(x + 4) \) is in point - slope form \( y - y_1=m(x - x_1) \), where the point \( (x_1,y_1)=(- 4,-6) \) and the slope \( m = \frac{3}{4} \).

Step2: Check the points on the graph

We can also find the y - intercept by setting \( x = 0 \) in the equation:

$$
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So the line should pass through the point \( (-4,-6) \) and \( (0,-3) \). Looking at the first graph, we can see that it has the point \( (-4,-6) \) and \( (0,-3) \) marked on it, while the second graph does not match these points. Also, we can check the slope. The slope \( m=\frac{3}{4} \), which means for a run of 4 (change in x), the rise is 3 (change in y). From the point \( (-4,-6) \), if we move 4 units to the right (x becomes \( - 4+4 = 0 \)) and 3 units up (y becomes \( -6 + 3=-3 \)), we reach the point \( (0,-3) \), which is consistent with the first graph.

Answer:

The graph with the points \((-4,-6)\) and \((0, - 3)\) (the upper graph)