Question

there is a coordinate grid with a line plotted on it. the x - axis and y - axis are labeled with numbers from - 6 to 6. the line passes through some grid points. there are some hand - written marks around the grid, but no clear question text.

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through the points \((0, -1)\) (y - intercept) and \((1, -2)\) (we can also use other points like \((5, -5)\) or \((1,0)\)? Wait, no, let's check the grid. Wait, the x - axis and y - axis: let's re - identify the points. Wait, looking at the graph, when \(x = 0\), \(y=- 1\)? Wait, no, maybe I misread. Wait, the line passes through \((0, - 1)\) and \((1,0)\)? No, wait, let's take two clear points. Let's take \((0,-1)\) and \((1,0)\)? Wait, no, the slope calculation: let's take two points. Let's see, when \(x = 0\), \(y=-1\) (the y - intercept), and when \(x = 1\), \(y = 0\)? No, that would be a positive slope, but the line is going down from left to right, so slope should be negative. Wait, maybe the points are \((0,-1)\) and \((1, - 2)\). Let's calculate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((x_1,y_1)=(0,-1)\) and \((x_2,y_2)=(1,-2)\). Then \(m=\frac{-2-(-1)}{1 - 0}=\frac{-1}{1}=-1\).

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

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We found \(m=-1\) and \(b=-1\) (since when \(x = 0\), \(y=-1\)). So the equation of the line is \(y=-x - 1\)? Wait, no, wait, let's check another point. Let's take \(x = 5\), then \(y=-5 - 1=-6\)? No, the point on the graph at \(x=-5\) has \(y = 4\)? Wait, maybe I misidentified the axes. Wait, the x - axis and y - axis: maybe the horizontal axis is x and vertical is y, but the labels are a bit confusing. Wait, let's re - examine the graph. Let's take two points: \((-5,4)\) and \((1,0)\). Now calculate the slope \(m=\frac{0 - 4}{1-(-5)}=\frac{-4}{6}=-\frac{2}{3}\)? No, that doesn't seem right. Wait, maybe the correct points are \((0,-1)\) and \((5,4)\)? No, the line passes through \((0,-1)\) and \((1,0)\) is wrong. Wait, let's use the standard method. Let's take two points: from the graph, the line passes through \((0,-1)\) and \((1,0)\) is incorrect. Wait, the line goes through \((0,-1)\) and \((1,0)\) would have a positive slope, but the line is decreasing, so slope is negative. Wait, maybe the points are \((0, - 1)\) and \((-1,0)\). Then slope \(m=\frac{0-(-1)}{-1 - 0}=\frac{1}{-1}=-1\). Ah, that makes sense. So if \(x=-1\), \(y = 0\) and \(x = 0\), \(y=-1\). Then slope \(m=-1\), y - intercept \(b=-1\). So the equation of the line is \(y=-x - 1\)? Wait, no, when \(x=-1\), \(y = 0\): \(0=-(-1)-1=1 - 1 = 0\), correct. When \(x = 0\), \(y=-0 - 1=-1\), correct. When \(x = 1\), \(y=-1 - 1=-2\), correct. So the equation of the line is \(y=-x - 1\). Wait, but maybe the question is to find the equation of the line. Let's confirm. The slope is \(-1\) and y - intercept is \(-1\), so \(y=-x - 1\).

Answer:

The equation of the line is \(y=-x - 1\)