Question

write the equation of this line in slope-intercept form.

Explanation:

Step1: Identify slope and intercept

Slope-intercept form is \( y = mx + b \), where \( m \) is slope and \( b \) is y-intercept. From the graph, the line crosses y-axis at \( (0, -3) \), so \( b = -3 \). To find slope, use two points. Let's take \( (0, -3) \) and \( (1, -8) \) (approximate, or see the rise over run). The change in \( y \) is \( -8 - (-3) = -5 \), change in \( x \) is \( 1 - 0 = 1 \), so slope \( m = -5 \). Wait, let's check another way. Wait, maybe better points. Wait, when \( x = 0 \), \( y = -3 \). When \( x = 1 \), \( y = -8 \)? Wait, no, maybe I misread. Wait, the line goes through (0, -3) and let's see, from (0, -3) to (1, -8)? Wait, no, maybe the slope is steeper. Wait, actually, looking at the graph, when \( x \) increases by 1, \( y \) decreases by 5? Wait, let's recalculate. Let's take two clear points. The y-intercept is at \( (0, -3) \). Let's take another point, say when \( x = 1 \), what's \( y \)? From the graph, the line at \( x = 1 \) is at \( y = -8 \)? Wait, no, maybe I made a mistake. Wait, actually, the slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points: (0, -3) and (1, -8). Then \( m = \frac{-8 - (-3)}{1 - 0} = \frac{-5}{1} = -5 \). Wait, but let's check another pair. If \( x = -1 \), what's \( y \)? At \( x = -1 \), the line is at \( y = 2 \)? Wait, no, the line at \( x = -1 \) is above the x-axis? Wait, maybe I misread the graph. Wait, the green line: when \( x = 0 \), it's at \( y = -3 \). When \( x = 1 \), it's at \( y = -8 \)? Wait, no, maybe the slope is -5? Wait, or maybe I made a mistake. Wait, let's look again. The y-intercept is at (0, -3). Let's take another point: when \( x = 1 \), the line is at \( y = -8 \)? So the slope is \( ( -8 - (-3) ) / (1 - 0) = -5 \). So the equation is \( y = -5x - 3 \)? Wait, no, wait, maybe the slope is -5? Wait, let's confirm. Slope-intercept form is \( y = mx + b \), where \( b \) is the y-intercept. So \( b = -3 \). Then, to find \( m \), we use two points. Let's take (0, -3) and (1, -8). Then \( m = (-8 - (-3))/(1 - 0) = -5/1 = -5 \). So the equation is \( y = -5x - 3 \)? Wait, but let's check with \( x = -1 \). If \( x = -1 \), then \( y = -5(-1) - 3 = 5 - 3 = 2 \). Looking at the graph, at \( x = -1 \), the line is at \( y = 2 \), which matches. So that works. So the slope is -5, y-intercept is -3. So the equation is \( y = -5x - 3 \). Wait, but let's check another point. At \( x = 2 \), \( y = -5(2) - 3 = -10 - 3 = -13 \), but the graph only goes to \( y = -8 \), so maybe my initial points were wrong. Wait, maybe I misread the y-intercept. Wait, the line crosses the y-axis at (0, -3)? Wait, no, looking at the graph, the y-axis is the vertical line. The line crosses the y-axis at (0, -3)? Wait, no, at \( x = 0 \), the line is at \( y = -3 \)? Wait, the grid lines: each grid is 1 unit. So the y-axis: from 0 down to -3, so the intercept is (0, -3). Then, when \( x = 1 \), the line is at \( y = -8 \)? Wait, no, the green line at \( x = 1 \) is at \( y = -8 \)? Wait, the bottom of the green line is at (1, -8)? So from (0, -3) to (1, -8), the change in y is -5, change in x is 1, so slope is -5. So the equation is \( y = -5x - 3 \). Wait, but let's confirm with \( x = -1 \): \( y = -5(-1) - 3 = 5 - 3 = 2 \), which is on the line (at \( x = -1 \), y = 2). Yes, that matches. So the slope is -5, y-intercept is -3. So the equation is \( y = -5x - 3 \).

Step2: Write the equation

Slope-intercept form is \( y = mx + b \), where \( m = -5 \) and \( b = -3 \). So substituting, we get \( y = -5x - 3 \).

Answer:

\( y = -5x - 3 \)