Question

1. $y = 3x - 5$
2. $y + 1 = 2(x + 3)$

Explanation:

Problem 20: Graph \( y = 3x - 5 \)

Step 1: Identify the slope and y-intercept

The equation is in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For \( y = 3x-5 \), the slope \( m = 3=\frac{3}{1} \) and the y - intercept \( b=- 5 \).

Step 2: Plot the y - intercept

The y - intercept is \( (0,-5) \). Locate the point \( (0,-5) \) on the coordinate plane.

Step 3: Use the slope to find another point

The slope \( \frac{3}{1} \) means we move up 3 units and then 1 unit to the right from the y - intercept. Starting from \( (0,-5) \), moving up 3 units gives \( y=-5 + 3=-2 \) and moving 1 unit to the right gives \( x = 0+1 = 1 \). So we get the point \( (1,-2) \). We can also move down 3 units and 1 unit to the left. Starting from \( (0,-5) \), moving down 3 units gives \( y=-5-3=-8 \) and moving 1 unit to the left gives \( x=0 - 1=-1 \), so we get the point \( (-1,-8) \).

Step 4: Draw the line

Draw a straight line through the points we have plotted (e.g., \( (0,-5) \), \( (1,-2) \), \( (-1,-8) \)).

Problem 22: Graph \( y + 1=2(x + 3) \)

Step 1: Rewrite in slope - intercept form (optional) or identify the point - slope form

The equation \( y - y_1=m(x - x_1) \) is the point - slope form, where \( (x_1,y_1) \) is a point on the line and \( m \) is the slope. For \( y + 1=2(x + 3) \), we can rewrite it as \( y-(-1)=2(x-(-3)) \). So the slope \( m = 2 \) and the line passes through the point \( (-3,-1) \).

Step 2: Plot the point

Locate the point \( (-3,-1) \) on the coordinate plane.

Step 3: Use the slope to find another point

The slope \( m = 2=\frac{2}{1} \). From the point \( (-3,-1) \), move up 2 units and 1 unit to the right. Moving up 2 units: \( y=-1 + 2 = 1 \), moving 1 unit to the right: \( x=-3 + 1=-2 \). So we get the point \( (-2,1) \). We can also move down 2 units and 1 unit to the left. From \( (-3,-1) \), moving down 2 units: \( y=-1-2=-3 \), moving 1 unit to the left: \( x=-3-1=-4 \), so we get the point \( (-4,-3) \).

Step 4: Draw the line

Draw a straight line through the points we have plotted (e.g., \( (-3,-1) \), \( (-2,1) \), \( (-4,-3) \)).

Answer:

For \( y = 3x-5 \): The line passes through \( (0,-5) \), \( (1,-2) \), \( (-1,-8) \) (among other points) and has a slope of 3.
For \( y + 1=2(x + 3) \): The line passes through \( (-3,-1) \), \( (-2,1) \), \( (-4,-3) \) (among other points) and has a slope of 2. The graph of \( y + 1=2(x + 3) \) can also be rewritten as \( y=2x + 5 \) (by expanding \( y+1 = 2x+6\Rightarrow y=2x + 5 \)), with y - intercept \( (0,5) \) and slope 2.