Question

How many solutions does the system of equations have?

y=−2x+12

4y+x−20=0

(1 point)
Responses

No solution
No solution

Infinitely many solutions
Infinitely many solutions

Two solutions
Two solutions

One solution

Answer 1

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

1. The first equation is in slope-intercept form:
\[
y = -2x + 12
\]
Here, the slope is -2 and the y-intercept is 12.

2. The second equation can be rearranged into slope-intercept form as well. Starting with:
\[
4y + x - 20 = 0
\]
we isolate \(y\):
\[
4y = -x + 20
\]
\[
y = -\frac{1}{4}x + 5
\]
In this equation, the slope is \(-\frac{1}{4}\) and the y-intercept is 5.

Now, we can analyze the slopes of the two lines:
- The slope of the first line is -2.
- The slope of the second line is \(-\frac{1}{4}\).

Since the slopes of the two lines are different, the two lines are not parallel and will intersect at one point.

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