Question

write the equation of the line in fully simplified slope - intercept form.

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((0, -1)\) (the y - intercept) and \((2, 2)\)? Wait, no, let's check again. Wait, the axes: the horizontal axis is x? Wait, no, looking at the labels, the vertical axis is x? Wait, no, the standard is x - axis horizontal, y - axis vertical. Wait, in the graph, the right - pointing arrow is labeled y, and the down - pointing arrow is labeled x? Wait, that's a bit confusing. Wait, let's re - orient. Let's take two clear points. Let's see, when x (the vertical axis) is 0, y (the horizontal axis) is - 1? Wait, no, maybe I got the axes reversed. Wait, the line passes through \((-1, 0)\) and \((2, - 3)\)? Wait, no, let's use the slope formula. The slope - intercept form is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Wait, let's find two points. Let's take the point where the line crosses the y - axis (if we consider the horizontal axis as y and vertical as x). Wait, the line passes through \((0, - 1)\) (if x = 0, y=-1) and \((3, - 4)\)? Wait, no, let's calculate the slope. Let's take two points: \((x_1,y_1)=(-1,0)\) and \((x_2,y_2)=(2, - 3)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-3 - 0}{2-(-1)}=\frac{-3}{3}=-1\). Wait, no, maybe I mixed up x and y. Wait, the correct way: let's assume the horizontal axis is y and vertical is x. Wait, the problem says "slope - intercept form" which is \(y = mx + b\), where y is the dependent variable (horizontal) and x is the independent variable (vertical)? No, standard is \(y=mx + b\) with x horizontal, y vertical. Wait, looking at the graph, the line goes from the second quadrant to the fourth quadrant. Let's take two points: when x (horizontal) is - 1, y (vertical) is 0? No, maybe the axes are labeled differently. Wait, the right - pointing arrow is y, up - down is x. So the line passes through (x = 0, y=-1) and (x = 3, y=-4). So the slope \(m=\frac{-4-(-1)}{3 - 0}=\frac{-3}{3}=-1\). And the y - intercept \(b=-1\) (when x = 0, y=-1). Wait, no, if x is the vertical axis and y is the horizontal axis, then the equation would be \(y=mx + b\) where y is horizontal, x is vertical. Wait, this is confusing. Wait, let's do it properly. Let's take two points with clear coordinates. Let's see, the line passes through (x = - 1, y = 0) and (x = 2, y=-3). So the change in y is \(-3-0=-3\), change in x is \(2-(-1)=3\), so slope \(m=\frac{-3}{3}=-1\). The y - intercept: when x = 0, what is y? Let's use the point - slope form. Using point \((-1,0)\) and \(m=-1\), the equation is \(y - 0=-1(x + 1)\), which simplifies to \(y=-x - 1\). Wait, let's check another point. If x = 2, y=-2 - 1=-3, which matches the point (2, - 3). If x=-1, y = 1 - 1 = 0, which matches (-1,0). So the slope is - 1 and the y - intercept is - 1.

Step2: Write the slope - intercept form

The slope - intercept form is \(y=mx + b\), where \(m=-1\) and \(b = - 1\). So the equation is \(y=-x-1\). Wait, let's verify with another point. When x = 3, y=-3 - 1=-4, which should be on the line. Yes, that works.

Answer:

\(y=-x - 1\)