How many solutions does the system of equations have?
y=−2x+12
4y+x−20=0
(1 point)
Responses
One solution
One solution
Two solutions
Two solutions
Infinitely many solutions
Infinitely many solutions
No solution
One solution
To determine the number of solutions for the system of equations, we can solve the system.
First, let's solve it using the substitution method.
We have the equation y = -2x + 12.
Substitute this value of y into the second equation:
4(-2x + 12) + x - 20 = 0
Simplify the equation:
-8x + 48 + x - 20 = 0
Simplify further:
-7x + 28 = 0
Add 7x to both sides of the equation:
-7x + 7x + 28 = 7x
28 = 7x
Divide both sides by 7:
4 = x
Now substitute this value of x back into the first equation:
y = -2(4) + 12
y = -8 + 12
y = 4
Therefore, the solution to the system of equations is x = 4 and y = 4.
Since there is only one set of values for x and y that satisfy both equations, the system has one solution.
So the answer is: One solution
To determine the number of solutions in a system of equations, we need to solve the equations simultaneously and check how many intersection points they have.
First, we have the equations:
y = -2x + 12
4y + x - 20 = 0
To solve this system of equations, we can use the substitution method or the elimination method. Here, I'll show you how to solve it using the elimination method:
Step 1: Multiply the first equation by 4 to make the x term have the same coefficient as the x term in the second equation:
4y = -8x + 48
Step 2: Now we can eliminate the x term by adding the modified first equation with the second equation:
4y + x - 20 = 0
-8x + 4y + 48 = 0
______________________
-7x + 28 = 0
Step 3: Solve the resulting equation for x:
-7x = -28
x = 4
Step 4: Substitute the value of x into one of the original equations (I'll use the first equation):
y = -2(4) + 12
y = -8 + 12
y = 4
After solving, we find that x = 4 and y = 4. Therefore, this system of equations has one solution.
So the answer is: One solution.