How many solutions does the system of equations have?%0D%0A%0D%0Ay=−2x+12%0D%0A%0D%0A=%0D%0A−%0D%0A2%0D%0A%0D%0A+%0D%0A12%0D%0A4y+x−20=0%0D%0A4%0D%0A%0D%0A+%0D%0A%0D%0A−%0D%0A20%0D%0A=%0D%0A0%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0AInfinitely many solutions%0D%0AInfinitely many solutions%0D%0A%0D%0ATwo solutions%0D%0ATwo solutions%0D%0A%0D%0ANo solution%0D%0ANo solution%0D%0A%0D%0AOne solution

Answer 1

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.