Question

How many solutions does the system of equations have?

y=−2x+12


4y+x−20=0


(1 point)
Responses

Two solutions
Two solutions

Infinitely many solutions
Infinitely many solutions

One solution
One solution

No solution

Answer 1

To determine how many solutions the system of equations has, we need to analyze the given equations:

1. \( y = -2x + 12 \) (This is in slope-intercept form.)
2. \( 4y + x - 20 = 0 \)

First, let's rewrite the second equation in slope-intercept form (solve for \( y \)):

Starting from \( 4y + x - 20 = 0 \), we can isolate \( y \):

\[
4y = -x + 20
\]

\[
y = -\frac{1}{4}x + 5
\]

Now, we have both equations:

1. \( y = -2x + 12 \)
2. \( y = -\frac{1}{4}x + 5 \)

Next, we examine the slopes of both equations:

- For the first equation, the slope is \(-2\).
- For the second equation, the slope is \(-\frac{1}{4}\).

Since the slopes are different, the lines represented by these equations will intersect at exactly one point.

Thus, the system of equations has **one solution**.

The correct response is:
**One solution**.