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The following system of linear equations is shown in the graph.

y equals one fourth times x plus 5
x − 4y = 4

a coordinate plane with one line that passes through the points 0 comma 5 and negative 4 comma 4 and another line that passes through the points 0 comma negative 1 and 4 comma 0

How many solutions does the system of linear equations have?
No solution
Infinitely many solutions
One solution at (4, 0)
One solution at (0, −1)

1 answer

To determine how many solutions the system of linear equations has, we need to analyze the equations given and their graphical representation.

We have the equations:

  1. \( y = \frac{1}{4}x + 5 \)
  2. \( x - 4y = 4 \)

Let's rewrite the second equation in slope-intercept form (y = mx + b):

\[ x - 4y = 4 \implies -4y = -x + 4 \implies y = \frac{1}{4}x - 1 \]

Now we have both equations in slope-intercept form:

  1. \( y = \frac{1}{4}x + 5 \)
  2. \( y = \frac{1}{4}x - 1 \)

Next, let's analyze their slopes and y-intercepts:

  • The first equation has a slope of \( \frac{1}{4} \) and a y-intercept of \( 5 \).
  • The second equation has a slope of \( \frac{1}{4} \) and a y-intercept of \( -1 \).

Since both equations have the same slope but different y-intercepts, it indicates that the lines are parallel and will never intersect.

In conclusion, the system of linear equations has:

No solution

This is because parallel lines do not intersect.