Question

How many solutions does the system of equations have? y=−2x+12 4y+x−20=0 (1 point) Responses Infinitely many solutions Infinitely many solutions One solution One solution Two solutions Two solutions No solution

Answer 1

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

The first equation is:
\[ y = -2x + 12 \]

The second equation can be rewritten in slope-intercept form (y = mx + b). The second equation is:
\[ 4y + x - 20 = 0 \]
Rearranging this gives:
\[ 4y = -x + 20 \]
\[ y = -\frac{1}{4}x + 5 \]

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

Next, we can determine the slopes of the two equations:
- The slope of the first equation (Equation 1) is -2.
- The slope of the second equation (Equation 2) is -1/4.

Since the slopes are different, the lines represented by these equations will intersect at one point, and thus the system has exactly **one solution**.

Therefore, the answer is:
**One solution**.