Question

find the equation of the graphed line.

a. ( y = -6x - 7 )
b. ( y = 6x - 7 )
c. ( y = -\frac{6}{7}x - 6 )
d. ( y = -\frac{7}{6}x - 7 )

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((-6, 1)\) and \((0, -7)\) (the y - intercept).

Step2: Calculate the slope \(m\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(-6,1)\) and \((x_2,y_2)=(0, - 7)\). Then \(m=\frac{-7 - 1}{0-(-6)}=\frac{-8}{6}=-\frac{4}{3}\)? Wait, no, maybe I misread the points. Wait, looking again, maybe the points are \((-6,1)\) and \((0, - 7)\)? Wait, no, let's check the grid again. Wait, the y - intercept is at \((0,-7)\), and another point: when \(x=-6\), \(y = 1\)? Wait, no, maybe I made a mistake. Wait, let's take two clear points. Let's see, when \(x = 0\), \(y=-7\) (so the y - intercept \(b=-7\)). Now, let's find another point. Let's take \(x=-6\), what's \(y\)? From the graph, when \(x=-6\), \(y = 1\)? Wait, no, maybe the slope is calculated as \(\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((x_1,y_1)=(-6,1)\) and \((x_2,y_2)=(0,-7)\). Then \(m=\frac{-7 - 1}{0+6}=\frac{-8}{6}=-\frac{4}{3}\)? But that's not one of the options. Wait, maybe I misread the points. Wait, the options have slopes like \(-\frac{7}{6}\), \(-\frac{6}{7}\), etc. Let's check the points again. Let's take two points: let's say \((-6,0)\) and \((0, - 7)\)? No, the x - intercept: when \(y = 0\), \(x=-6\)? Wait, the line crosses the x - axis at \(x=-6\)? Wait, no, looking at the graph, the line passes through \((-6,1)\)? No, maybe the correct points are \((-6,0)\) and \((0,-7)\)? Wait, no, let's calculate the slope between \((-6,1)\) and \((0,-7)\): \(m=\frac{-7 - 1}{0 - (-6)}=\frac{-8}{6}=-\frac{4}{3}\), which is not in the options. Wait, maybe the points are \((-6,1)\) and \((0,-7)\) is wrong. Wait, let's check the options. The options are:

a. \(y=-6x - 7\) (slope - 6)

b. \(y = 6x-7\) (slope 6)

c. \(y=-\frac{6}{7}x-6\) (slope \(-\frac{6}{7}\), y - intercept - 6)

d. \(y=-\frac{7}{6}x-7\) (slope \(-\frac{7}{6}\), y - intercept - 7)

Ah! I see, I misread the y - intercept. The y - intercept is at \((0,-7)\), so \(b = - 7\). Now, let's find the slope between two points. Let's take \((x_1,y_1)=(-6,0)\) and \((x_2,y_2)=(0,-7)\). Wait, no, when \(y = 0\), \(x=-6\)? Wait, no, if \(x=-6\), \(y = 0\) (x - intercept) and \(x = 0\), \(y=-7\) (y - intercept). Then the slope \(m=\frac{-7-0}{0 - (-6)}=\frac{-7}{6}\). Yes! That's the slope. So the slope \(m=-\frac{7}{6}\) and the y - intercept \(b=-7\). So the equation is \(y=-\frac{7}{6}x-7\), which is option d.

Answer:

d. \(y =-\frac{7}{6}x-7\)