Question

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

Explanation:

Step1: Analyze the slopes

The two equations are \( y = \frac{1}{3}x + 3 \) and \( y = \frac{1}{3}x - 2 \). The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, both lines have the same slope \( m=\frac{1}{3} \).

Step2: Analyze the y-intercepts

For the first line, the y-intercept \( b = 3 \), so it crosses the y-axis at \( (0, 3) \). For the second line, the y-intercept \( b=-2 \), so it crosses the y-axis at \( (0, -2) \).

Step3: Determine the relationship

Since the slopes are equal and the y-intercepts are different, the two lines are parallel. To graph them, we can plot the y-intercepts and then use the slope (rise over run, \( \frac{1}{3} \) means rise 1, run 3) to draw the lines. The first line starts at \( (0, 3) \) and goes up 1, right 3 (or down 1, left 3). The second line starts at \( (0, -2) \) and follows the same slope.

Answer:

The two lines \( y = \frac{1}{3}x + 3 \) and \( y = \frac{1}{3}x - 2 \) are parallel. To graph them, plot \( (0, 3) \) for the first line and use the slope \( \frac{1}{3} \) to draw it, and plot \( (0, -2) \) for the second line and use the same slope to draw it.