Describe and correct the error a student made when graphing \( y+5=-\frac{3}{4}(x-8) \)

To help you solve the problem, let's first identify the error in the student's graphing process and then correct it. ### Error Identification: 1. **Incorrect Equation Transformation**: The student might have incorrectly transformed the given equation into the slope-intercept form \( y = mx + b \) or the standard form \( Ax + By = C \). 2. **Incorrect Slope and Y-Intercept**: If the student transformed the equation, they might have made an error in identifying the slope \( m \) and the y-intercept \( b \). 3. **Incorrect Plotting of Points**: The student might have plotted the points incorrectly on the graph. ### Correcting the Error: Let's start by transforming the given equation into the slope-intercept form: Given equation: \[ y + 5 = -\frac{3}{4}(x - 8) \] First, distribute the slope \(-\frac{3}{4}\) to the terms inside the parentheses: \[ y + 5 = -\frac{3}{4}x + 6 \] Now, isolate \( y \) by subtracting 5 from both sides: \[ y = -\frac{3}{4}x + 6 - 5 \] \[ y = -\frac{3}{4}x + 1 \] This is the equation in slope-intercept form, where: - The slope \( m \) is \(-\frac{3}{4}\) - The y-intercept \( b \) is \( 1 \) ### Steps to Correct the Graph: 1. **Plot the Y-Intercept**: Start by plotting the y-intercept, which is the point where the line crosses the y-axis. In this case, the y-intercept is \( (0, 1) \). 2. **Use the Slope to Plot Another Point**: The slope \(-\frac{3}{4}\) means for every 4 units you move up (positive y-direction), you move 3 units to the left (negative x-direction). Starting from the y-intercept \( (0, 1) \), move 3 units to the left and 4 units up to find another point. 3. **Draw the Line**: Connect the two points with a straight line to represent the graph of the equation. By following these steps, the student can correct the graph and ensure it accurately represents the equation \( y = -\frac{3}{4}x + 1 \).