Question

12 graph the given system of linear equations.
$y = -1$
$y = -\frac{5}{2}x + 4$
choose the correct line to graph: line 1 is line ab. line 2 is line cd
click to select
line ab
line cd

Explanation:

Step1: Analyze \( y = -1 \)

The equation \( y = -1 \) is a horizontal line (slope \( m = 0 \)) where all points have a \( y \)-coordinate of \( -1 \). So it's a horizontal line passing through \( (x, -1) \) for any \( x \).

Step2: Analyze \( y = -\frac{5}{2}x + 4 \)

This is in slope - intercept form \( y=mx + b \), where slope \( m=-\frac{5}{2} \) and \( y \)-intercept \( b = 4 \). So it crosses the \( y \)-axis at \( (0,4) \) and has a negative slope (decreasing line).

Step3: Match to Lines

• Line \( y=-1 \): Since it's horizontal at \( y=-1 \), we can assign it to one of the lines (say Line AB or Line CD). The line \( y = -\frac{5}{2}x+4 \) has a \( y \)-intercept at \( (0,4) \) and a slope of \( -\frac{5}{2} \) (for every 2 units right, it goes down 5 units).

To graph \( y=-1 \): It's a horizontal line through \( y=-1 \), so we can plot points like \( (-2,-1) \), \( (0,-1) \), \( (2,-1) \) etc.

To graph \( y = -\frac{5}{2}x + 4 \): Start at \( (0,4) \) (the \( y \)-intercept). Then from \( (0,4) \), move 2 units to the right (to \( x = 2 \)) and 5 units down (to \( y=4 - 5=-1 \)), so we get the point \( (2,-1) \). We can also move 2 units to the left (to \( x=-2 \)) and 5 units up (to \( y = 4+5 = 9 \)), getting the point \( (-2,9) \).

Answer:

To graph \( y=-1 \) (a horizontal line at \( y = - 1 \)) and \( y=-\frac{5}{2}x + 4 \) (a line with \( y \)-intercept \( (0,4) \) and slope \( -\frac{5}{2} \)):

• For \( y=-1 \): Plot points with \( y=-1 \) (e.g., \( (0,-1) \), \( (3,-1) \)) and draw a horizontal line.
• For \( y = -\frac{5}{2}x+4 \): Plot \( (0,4) \), then use the slope to find another point (e.g., from \( (0,4) \), move 2 right and 5 down to \( (2,-1) \)) and draw the line through these points.

If we consider Line 1 (Line AB) and Line 2 (Line CD):

• If Line AB is \( y=-1 \) (horizontal at \( y=-1 \)) and Line CD is \( y = -\frac{5}{2}x + 4 \) (with \( y \)-intercept \( (0,4) \) and passing through \( (2,-1) \)) or vice - versa, we can identify the correct lines by their characteristics.

The key is to recognize that \( y=-1 \) is horizontal and \( y = -\frac{5}{2}x + 4 \) is a non - horizontal line with a negative slope and \( y \)-intercept at \( (0,4) \).