To determine how many solutions the system of equations has, we can analyze the two equations:
-
The first equation is given in slope-intercept form: \[ y = -2x + 12 \]
-
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.