Question

How many solutions does the system of equations have?

y=−2x+12

4y+x−20=0

(1 point)
Responses

No solution
No solution

Two solutions
Two solutions

One solution
One solution

Infinitely many solutions
Infinitely many solutions

Answer 1

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

1. The first equation is in slope-intercept form:
\( y = -2x + 12 \)

2. The second equation can be rearranged into slope-intercept form. Starting with:
\( 4y + x - 20 = 0 \)
Rearranging gives:
\( 4y = -x + 20 \)
Dividing everything by 4:
\( y = -\frac{1}{4}x + 5 \)

Now we have:

- First equation: \( y = -2x + 12 \)
- Second equation: \( y = -\frac{1}{4}x + 5 \)

Next, we compare the slopes of the two equations:
- The slope of the first equation is \(-2\).
- The slope of the second equation 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**.