Question

the image shows a coordinate grid with labels: origin, x-axis, y-axis, y-intercept (red dot), x-intercept (red dot), a red line, and green rectangles with arrows pointing to gray dots.

Answer:

To determine the equation of the line, we first identify the intercepts. From the graph:

Step 1: Identify Intercepts

• The y-intercept (\(b\)) is where the line crosses the \(y\)-axis. From the graph, the \(y\)-intercept is at \((0, -3)\)? Wait, no—wait, looking again: Wait, the red dot for \(y\)-intercept: Wait, the grid: Let's check coordinates. Wait, the \(y\)-intercept (red dot) is at \(x=0\), \(y=-3\)? No, wait, maybe I misread. Wait, the \(x\)-intercept (red dot) is at \(x=-1\)? Wait, no, let's re-express. Wait, the \(y\)-intercept (red circle) is at \((-3, 0)\)? No, no—wait, the \(y\)-axis is vertical, \(x\)-axis horizontal. Wait, the \(y\)-intercept is where \(x=0\). Looking at the graph: The red dot labeled "y-intercept" is at \(x=-3\), \(y=0\)? No, that's on the \(x\)-axis. Wait, no—wait, the "x-intercept" red dot is at \(x=-1\), \(y=0\)? Wait, I think I mixed up. Let's correct:

• y-intercept (\(b\)): The line crosses the \(y\)-axis at \((0, -3)\)? No, wait, the red dot for "y-intercept" is at \(x=-3\), \(y=0\)—that's on the \(x\)-axis. Wait, no, the labels: "y-intercept" is on the \(x\)-axis? No, that must be a mistake. Wait, no—wait, the graph: The red line crosses the \(x\)-axis at \((-1, 0)\) (x-intercept) and the \(y\)-axis at \((0, -3)\)? Wait, no, let's count the grid. Let's assume each grid square is 1 unit.

Looking at the "y-intercept" red dot: It's at \(x = -3\), \(y = 0\) (on the \(x\)-axis). Wait, that can't be. Wait, maybe the labels are swapped? No, the "x-intercept" is where \(y=0\), "y-intercept" where \(x=0\). Wait, the red dot labeled "y-intercept" is at \(x = -3\), \(y = 0\) (so that's actually the \(x\)-intercept). Wait, no, the other red dot is labeled "x-intercept" at \(x = -1\), \(y = 0\)? No, this is confusing. Wait, let's start over.

Wait, the line is red. Let's find two points. Let's take the \(y\)-intercept: when \(x=0\), what's \(y\)? Wait, the red dot labeled "y-intercept" is at \(x = -3\), \(y = 0\) (so that's \((-3, 0)\)). The red dot labeled "x-intercept" is at \(x = -1\), \(y = 0\)? No, that's on the \(x\)-axis. Wait, no—wait, the line crosses the \(y\)-axis at \((0, -3)\) and the \(x\)-axis at \((-3, 0)\)? Wait, no, let's calculate slope.

Wait, maybe the correct intercepts are:

• \(y\)-intercept (\(b\)): When \(x=0\), \(y = -3\) (so \((0, -3)\))? No, the red dot labeled "y-intercept" is at \(x = -3\), \(y = 0\) (so \((-3, 0)\) is on the \(x\)-axis, so that's the \(x\)-intercept). Wait, the label "y-intercept" is misplaced? Or maybe I misread. Let's assume:

• \(x\)-intercept: \((-3, 0)\) (where \(y=0\))
• \(y\)-intercept: \((0, -3)\) (where \(x=0\))

Then, the slope \(m\) is \(\frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 0}{0 - (-3)} = \frac{-3}{3} = -1\).

So the equation of the line in slope-intercept form (\(y = mx + b\)) is \(y = -x - 3\).

Wait, but maybe the intercepts are different. Let's check the graph again. The red line: when \(x = -3\), \(y = 0\) (x-intercept), and when \(x = 0\), \(y = -3\) (y-intercept). So slope \(m = \frac{-3 - 0}{0 - (-3)} = -1\). So equation is \(y = -x - 3\).

Alternatively, if the \(x\)-intercept is \((-1, 0)\) and \(y\)-intercept is \((0, -1)\), but that doesn't match. Wait, the grid: each vertical and horizontal line is 1 unit. So the "y-intercept" red dot is at \(x = -3\), \(y = 0\) (so \(x\)-intercept), and "x-intercept" red dot is at \(x = -1\), \(y = 0\)? No, that's on the same axis. I think there's a label error. Let's proceed with the visible points.

Wait, the line passes through \((-3, 0)\) and \((0, -3)\). So:

Step 1: Find Slope (\(m\))

Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Let \((x_1, y_1) = (-3, 0)\) and \((x_2, y_2) = (0, -3)\)
\(m = \frac{-3 - 0}{0 - (-3)} = \frac{-3}{3} = -1\)

Step 2: Find \(y\)-intercept (\(b\))

The \(y\)-intercept is where \(x = 0\). From the point \((0, -3)\), \(b = -3\).

Step 3: Write Equation

Slope-intercept form: \(y = mx + b\)
Substitute \(m = -1\) and \(b = -3\):
\(y = -x - 3\)

So the equation of the line is \(y = -x - 3\).