Question

Question
How many solutions does the system of equations have?

y=−2x+12

4y+x−20=0

(1 point)
Responses

Two solutions
Two solutions

One solution
One solution

Infinitely many solutions
Infinitely many solutions

No solution
No solution
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Answer 1

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

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

2. The second equation can be rearranged:
\[
4y + x - 20 = 0 \implies 4y = -x + 20 \implies y = -\frac{1}{4}x + 5
\]

Now we have:
- Equation 1: \(y = -2x + 12\)
- Equation 2: \(y = -\frac{1}{4}x + 5\)

Next, let's compare the slopes and y-intercepts of both lines:

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

Since the slopes are different, it means that the lines intersect at exactly one point. Therefore, there is only one solution to the system of equations.

The correct answer is **One solution**.