Question

graph the line.
$y + 2 = 5(x - 3)$

Explanation:

Step1: Identify the form of the equation

The given equation is \( y + 2 = 5(x - 3) \), which 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. Here, \( m = 5 \) and the point \( (x_1,y_1)=(3,-2) \).

Step2: Plot the point

Locate the point \( (3,-2) \) on the coordinate plane. Start at the origin, move 3 units to the right along the x - axis and then 2 units down along the y - axis.

Step3: Use the slope to find another point

The slope \( m = 5=\frac{5}{1} \), which means for a run of 1 unit (move 1 unit to the right along the x - axis), we have a rise of 5 units (move 5 units up along the y - axis). From the point \( (3,-2) \), moving 1 unit to the right (to \( x = 3+1 = 4 \)) and 5 units up (to \( y=-2 + 5=3 \)), we get the point \( (4,3) \). We can also use a run of - 1 (move 1 unit to the left) and a rise of - 5 (move 5 units down). From \( (3,-2) \), moving 1 unit to the left (to \( x = 3-1 = 2 \)) and 5 units down (to \( y=-2-5=-7 \)) gives the point \( (2,-7) \).

Step4: Draw the line

Draw a straight line passing through the points we found (e.g., \( (3,-2) \) and \( (4,3) \) or \( (3,-2) \) and \( (2,-7) \)).

Answer:

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

1. Plot the point \( (3,-2) \) (from the point - slope form \( y - y_1=m(x - x_1) \) where \( x_1 = 3,y_1=-2 \) and \( m = 5 \)).
2. Use the slope \( m = 5 \) (rise = 5, run = 1) to find another point. For example, from \( (3,-2) \), moving 1 unit right and 5 units up gives \( (4,3) \).
3. Draw a straight line through the plotted points.