Question

1. graph the equation below.

$y = -\frac{2}{3}x + 1$

1. solve and graph the equation below.

Explanation:

Step1: Identify the slope and y-intercept

The equation \( y = -\frac{2}{3}x + 1 \) is in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. Here, \( m=-\frac{2}{3} \) and \( b = 1 \).

Step2: Plot the y - intercept

The y - intercept \( b = 1 \) means the line crosses the y - axis at the point \( (0,1) \). Locate the point \( (0,1) \) on the coordinate plane.

Step3: Use the slope to find another point

The slope \( m=-\frac{2}{3}=\frac{\text{rise}}{\text{run}} \). The negative sign means we can go down 2 units (rise) and then to the right 3 units (run) from the y - intercept point \( (0,1) \). So, from \( (0,1) \), moving down 2 units gives \( y=1 - 2=-1 \) and moving right 3 units gives \( x = 0+3 = 3 \). So we get the point \( (3,-1) \). We can also go up 2 units and left 3 units (since \( \frac{- 2}{3}=\frac{2}{-3} \)). From \( (0,1) \), moving up 2 units gives \( y = 1+2 = 3 \) and moving left 3 units gives \( x=0 - 3=-3 \), so we get the point \( (-3,3) \).

Step4: Draw the line

Draw a straight line through the points \( (0,1) \), \( (3,-1) \) (or \( (-3,3) \)) to graph the equation \( y=-\frac{2}{3}x + 1 \).

Answer:

To graph \( y = -\frac{2}{3}x+1 \):

1. Plot the y - intercept at \( (0,1) \).
2. Use the slope \( -\frac{2}{3} \) to find another point (e.g., \( (3, - 1) \) or \( (-3,3) \)).
3. Draw a straight line through the plotted points. The line should pass through \( (0,1) \) and have a slope of \( -\frac{2}{3} \), going down 2 and right 3 (or up 2 and left 3) from the y - intercept.