To determine the number of solutions, we need to solve the system of equations.
The first equation is in slope-intercept form y = -2x + 12, which means it represents a line.
The second equation is 4y + x - 20 = 0, which can be rearranged to x = -4y + 20. This equation is also in slope-intercept form, representing a line.
To find the point of intersection, we can set the two equations equal to each other:
-2x + 12 = -4y + 20
Simplifying, we get:
-2x = -4y + 8
Dividing by -2, we have:
x = 2y - 4
Now we have a system of two equations:
y = -2x + 12
x = 2y - 4
Substituting the value of x from the second equation into the first equation, we get:
y = -2(2y - 4) + 12
Expanding and simplifying:
y = -4y + 8 + 12
Combining like terms:
5y = 20
Dividing by 5:
y = 4
Substituting the value of y into the second equation, we get:
x = 2(4) - 4
x = 8 - 4
x = 4
So the system of equations has one solution where x = 4 and y = 4.
Therefore, the correct answer is: One solution.