Question

11 of 13 aa.7 write a linear equation from a graph wyom question write the equation of this line in slope - intercept form. write your answer using integers, proper fractions, and improper fractions in simplest form. correct answer

Explanation:

Step1: Identify slope-intercept form

Slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step2: Find the y-intercept (\( b \))

The line crosses the y-axis at \( (0, 3) \), so \( b = 3 \).

Step3: Calculate the slope (\( m \))

Use two points, e.g., \( (0, 3) \) and \( (3, 0) \). Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{3 - 0} = \frac{-3}{3} = -1 \)? Wait, no, wait. Wait, another point: when \( x = 4 \), \( y = -1 \)? Wait, no, let's check the graph again. Wait, the line goes through \( (0, 3) \) and \( (3, 0) \)? Wait, no, when \( x = 3 \), \( y = 0 \)? Wait, no, the x-intercept is at \( x = 3 \)? Wait, no, looking at the graph, the line crosses the x-axis at \( x = 3 \)? Wait, no, the grid: from \( (0,3) \) to \( (3,0) \): change in y is \( 0 - 3 = -3 \), change in x is \( 3 - 0 = 3 \), so slope \( m = \frac{-3}{3} = -1 \)? Wait, no, wait, when x increases by 3, y decreases by 3, so slope is -1? Wait, but let's check another point. From \( (0,3) \) to \( (4, -1) \): change in y is \( -1 - 3 = -4 \), change in x is \( 4 - 0 = 4 \), so slope is \( \frac{-4}{4} = -1 \). Wait, no, wait, maybe I made a mistake. Wait, the y-intercept is 3, so \( b = 3 \). Then, let's take two points: (0, 3) and (3, 0). So slope \( m = \frac{0 - 3}{3 - 0} = -1 \). Wait, but let's check the equation. If \( y = -x + 3 \), then when x = 0, y = 3 (correct). When x = 3, y = 0 (correct). When x = 4, y = -1 (correct, as per the graph). So yes, slope is -1, y-intercept is 3. Wait, but wait, maybe I miscalculated. Wait, another way: the slope-intercept form is \( y = mx + b \). We know \( b = 3 \). Let's find the slope between (0,3) and (4, -1): \( m = \frac{-1 - 3}{4 - 0} = \frac{-4}{4} = -1 \). So slope is -1. So the equation is \( y = -x + 3 \)? Wait, but let's check again. Wait, the line: from (0,3) going down 1, right 1: (1,2), (2,1), (3,0), (4,-1), (5,-2), etc. Yes, that's correct. So slope is -1, y-intercept is 3. So the equation is \( y = -x + 3 \)? Wait, but wait, maybe I made a mistake. Wait, no, let's recalculate the slope. Wait, the formula for slope is \( m = \frac{\Delta y}{\Delta x} \). Let's take two points: (0, 3) and (3, 0). \( \Delta y = 0 - 3 = -3 \), \( \Delta x = 3 - 0 = 3 \), so \( m = \frac{-3}{3} = -1 \). Correct. So the equation is \( y = -x + 3 \). Wait, but wait, the problem says "Write the equation of this line in slope-intercept form." So slope-intercept form is \( y = mx + b \), where \( m \) is slope and \( b \) is y-intercept. So we found \( m = -1 \) and \( b = 3 \), so the equation is \( y = -x + 3 \). Wait, but let's check again. Wait, maybe the slope is \( -\frac{3}{4} \)? No, wait, no, the points: (0,3) and (4, -1): \( \Delta y = -4 \), \( \Delta x = 4 \), so slope is -1. Wait, no, 0 - 3 is -3, 3 - 0 is 3, so -3/3 is -1. Yes. So the equation is \( y = -x + 3 \). Wait, but let's confirm with the graph. The line goes through (0,3), (1,2), (2,1), (3,0), (4,-1), etc. So yes, that's a slope of -1. So the equation is \( y = -x + 3 \). Wait, but maybe I made a mistake in the slope. Wait, another approach: the general form of a line is \( y = mx + b \). We know that when x=0, y=3, so b=3. Then, we can pick another point, say (3,0), plug into the equation: 0 = m*3 + 3 → 3m = -3 → m = -1. So yes, m = -1. So the equation is \( y = -x + 3 \).

Wait, but wait, the user's graph: let's check again. The y-axis is vertical, x-axis horizontal. The line starts at the top left, goes through (0,3), then (3,0), (4,-1), etc. So the slope is -1, y-intercept 3. So the equation is…

Answer:

\( y = -x + 3 \)