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
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.
To determine the number of solutions for the system of equations, let's start by rewriting the equations:
1) y = -2x + 12
2) 4y + x - 20 = 0
We can solve this system of equations by substitution or elimination method. Let's use the elimination method:
First, let's multiply the second equation by 2 to make the coefficient of x the same as in the first equation:
2(4y + x - 20) = 2(0)
8y + 2x - 40 = 0
Now, we have:
1) y = -2x + 12
3) 8y + 2x - 40 = 0
Next, let's add equation 1 and equation 3:
(y) + (8y) = (-2x + 12) + (2x - 40)
9y = -28
Divide both sides of the equation by 9:
y = -28/9
Substitute the value of y into equation 1:
-28/9 = -2x + 12
Multiply both sides by 9 to eliminate the fraction:
-28 = -18x + 108
Simplify the equation:
-18x = -136
Divide both sides by -18:
x = 136/18
x = 68/9
Therefore, the system of equations has one solution.
To determine the number of solutions for the given system of equations, let's start by analyzing the equations:
Equation 1: y = -2x + 12
Equation 2: 4y + x - 20 = 0
To find the number of solutions, we can solve these equations using the method that is most suitable in this case - substitution, elimination, or graphing. Let's use the substitution method:
1. Solve Equation 1 for y in terms of x:
y = -2x + 12
2. Substitute this expression into Equation 2 for y:
4(-2x + 12) + x - 20 = 0
3. Simplify and solve for x:
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = 4
4. Substitute the value of x back into Equation 1 to find y:
y = -2(4) + 12
y = -8 + 12
y = 4
After solving, we found that x = 4 and y = 4. This means that the given system of equations has one unique solution.
Therefore, the correct answer is: One solution.