Question

graph the line.
$y - 5 = \frac{2}{3}(x + 4)$

Explanation:

Step1: Identify the form of the equation

The given equation is \( y - 5=\frac{2}{3}(x + 4) \), which is in the point - slope form \( y - y_1=m(x - x_1) \), where \( m=\frac{2}{3} \) (the slope) and the point \( (x_1,y_1)=(- 4,5) \).

Step2: Plot the point

First, we plot the point \( (-4,5) \) on the coordinate plane. To find another point, we use the slope. The slope \( m = \frac{2}{3}\) means that for a run of 3 units (increase in \( x \) by 3), the rise is 2 units (increase in \( y \) by 2). So, starting from the point \( (-4,5) \), if we add 3 to the \( x \) - coordinate (\( - 4+3=-1 \)) and add 2 to the \( y \) - coordinate (\( 5 + 2=7 \)), we get the point \( (-1,7) \). We can also go in the opposite direction: subtract 3 from the \( x \) - coordinate (\( -4-3=-7 \)) and subtract 2 from the \( y \) - coordinate (\( 5-2 = 3 \)) to get the point \( (-7,3) \).

Step3: Draw the line

After plotting at least two points (e.g., \( (-4,5) \), \( (-1,7) \) or \( (-4,5) \), \( (-7,3) \)), we draw a straight line passing through these points.

To graph the line \( y - 5=\frac{2}{3}(x + 4) \):

1. Plot the point \( (-4,5) \) (since from \( y - y_1=m(x - x_1) \), \( x_1=-4,y_1 = 5 \)).
2. Use the slope \( m=\frac{2}{3} \):

• From \( (-4,5) \), move 3 units to the right (increase \( x \) by 3: \( - 4+3=-1 \)) and 2 units up (increase \( y \) by 2: \( 5 + 2 = 7 \)) to get the point \( (-1,7) \).
• Or move 3 units to the left (decrease \( x \) by 3: \( -4-3=-7 \)) and 2 units down (decrease \( y \) by 2: \( 5-2 = 3 \)) to get the point \( (-7,3) \).

1. Draw a straight line through the plotted points.

Answer:

The line is graphed by plotting the point \((-4,5)\) and using the slope \(\frac{2}{3}\) to find additional points (e.g., \((-1,7)\) or \((-7,3)\)) and then drawing a straight line through them.