Consider the system of equations shown below.
[tex]
\begin{array}{l}
y=-5 x+1 \
y=-5 x+10
\end{array}
[/tex]
When graphed, the system consists of two lines that will never meet, no matter how far they are extended. Why are the lines parallel?
A. The linear equations have the same slope and [tex]$y$[/tex]-intercept.
B. The linear equations have different slopes and [tex]$y$[/tex]-intercepts.
C. The linear equations have the same slope but different [tex]$y$[/tex]-intercepts.
D. The linear equations have different slopes but the same [tex]$y$[/tex]-intercept.
Answer (2)
The equations are y = − 5 x + 1 and y = − 5 x + 10 .
Both lines have a slope of -5.
The y-intercepts are 1 and 10, respectively.
Therefore, the lines are parallel because they have the same slope but different y-intercepts. The answer is: The linear equations have the same slope but different y -intercepts.
Explanation
Analyze the equations Let's analyze the given system of equations:
Equation 1: y = − 5 x + 1 Equation 2: y = − 5 x + 10
We need to determine why these lines are parallel. Recall that parallel lines have the same slope but different y-intercepts. Let's identify the slope and y-intercept for each equation.
Identify slopes and y-intercepts For Equation 1, y = − 5 x + 1 , the slope is -5 and the y-intercept is 1. For Equation 2, y = − 5 x + 10 , the slope is -5 and the y-intercept is 10.
Compare and conclude Comparing the slopes and y-intercepts, we see that both lines have the same slope (-5), but different y-intercepts (1 and 10). Therefore, the lines are parallel because they have the same slope but different y-intercepts.
Examples
Imagine you're designing a staircase where each step is represented by a line on a graph. If two sets of steps have the same steepness (slope) but start at different heights (y-intercepts), they will run parallel to each other and never meet. This concept applies in various fields, such as architecture, where understanding parallel lines ensures structures are stable and aesthetically pleasing, or in urban planning, where parallel roads can optimize traffic flow and prevent intersections.
The lines are parallel because they have the same slope of -5, but different y-intercepts of 1 and 10. This means they will never meet. The correct option is C.
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