Question

graph the line $y + 2 = \frac{2}{3}(x + 3)$.

Explanation:

Step1: Identify the form of the line

The equation \( y + 2=\frac{2}{3}(x + 3) \) is in point - slope form \( y - y_1=m(x - x_1) \), where \( m \) is the slope and \( (x_1,y_1) \) is a point on the line. By comparing, we have \( m = \frac{2}{3} \) and the point \( (x_1,y_1)=(-3,-2) \).

Step2: Plot the point

First, we plot the point \( (-3,-2) \) on the coordinate plane.

Step3: Use the slope to find another point

The slope \( m=\frac{2}{3} \) means that for a run (change in \( x \)) of 3 units, the rise (change in \( y \)) is 2 units. Starting from the point \( (-3,-2) \), if we move 3 units to the right (increase \( x \) by 3: \( - 3+3 = 0 \)) and 2 units up (increase \( y \) by 2: \( -2 + 2=0 \)), we get the point \( (0,0) \).

Step4: Draw the line

Now we can draw a straight line passing through the points \( (-3,-2) \) and \( (0,0) \) (and we can find more points using the slope if needed) to graph the line.

Answer:

To graph the line \( y + 2=\frac{2}{3}(x + 3) \):

1. Plot the point \( (-3,-2) \) (from the point - slope form \( y - y_1=m(x - x_1) \) where \( x_1=-3,y_1 = - 2 \)).
2. Use the slope \( m=\frac{2}{3} \): from \( (-3,-2) \), move 3 units right and 2 units up to get the point \( (0,0) \).
3. Draw a straight line through the points \( (-3,-2) \) and \( (0,0) \) (and extend it in both directions).