Question

graph the following linear equations using the slope
a) $y = \frac{-3}{2}x + 7$

Explanation:

Step1: Identify Slope-Intercept Form

The equation \( y = \frac{-3}{2}x + 7 \) is in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. Here, \( m=-\frac{3}{2} \) and \( b = 7 \).

Step2: Locate the Y - Intercept

The y - intercept \( b = 7 \) means the line crosses the y - axis at the point \( (0,7) \). Looking at the graph, we can see that the line intersects the y - axis at \( (0,7) \), which matches the y - intercept from the equation.

Step3: Use the Slope to Find Another Point

The slope \( m=-\frac{3}{2} \) can be interpreted as "rise over run", or \( \frac{\text{change in }y}{\text{change in }x} \). A slope of \( -\frac{3}{2} \) means that for a run (change in \( x \)) of \( 2 \) units, the rise (change in \( y \)) is \( - 3 \) units (or for a run of \( - 2 \) units, the rise is \( 3 \) units). Starting from the y - intercept \( (0,7) \), if we move \( 2 \) units to the right (increase \( x \) by \( 2 \)) and \( 3 \) units down (decrease \( y \) by \( 3 \)), we get the point \( (0 + 2,7-3)=(2,4) \). We can see that this point lies on the given graph. Repeating this process, moving \( 2 \) units right and \( 3 \) units down from \( (2,4) \) gives \( (4,1) \), and so on. Also, moving in the opposite direction (left \( 2 \), up \( 3 \)) from \( (0,7) \) gives \( (-2,10) \), which should also lie on the line (if we were to extend it). The graph provided has a line that passes through these types of points, consistent with the slope - intercept form of the equation \( y =-\frac{3}{2}x + 7 \). So, the graph is a correct representation of the linear equation \( y =-\frac{3}{2}x + 7 \).

Answer:

The graph is a correct representation of the linear equation \( y=\frac{-3}{2}x + 7 \) (since it passes through the y - intercept \( (0,7) \) and follows the slope \( -\frac{3}{2} \) as shown by the pattern of the points on the line).