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

Infinitely many solutions
Infinitely many solutions

One solution
One solution

No solution
No solution
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Answer 1

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

1. \( y = -2x + 12 \) (This is the equation of a line in slope-intercept form.)

2. \( 4y + x - 20 = 0 \) (We can rearrange this to get it in slope-intercept form as well.)

Let's rearrange the second equation:

\[ 4y = -x + 20 \] \[ y = -\frac{1}{4}x + 5 \]

Now we have:

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

Next, we compare the slopes of both lines:

• The slope of the first line (\( y = -2x + 12 \)) is \(-2\).
• The slope of the second line (\( y = -\frac{1}{4}x + 5 \)) is \(-\frac{1}{4}\).

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

Thus, the system of equations has one solution.