How many solutions does the system of equations have?
y = −2x +12
4y + x −20 = 0
Responses
No Solution
No Solution
Infinitely many solutions
Infinitely many solutions
One solution
One solution
Two solutions
Two solutions
One solution
To determine the number of solutions, we can solve the system of equations. Let's start by solving them:
First, let's rearrange the second equation to isolate one variable:
4y + x - 20 = 0
Subtract x from both sides:
4y = 20 - x
Divide by 4:
y = (20 - x)/4
Now we have the value of y in terms of x. We can substitute this expression for y into the first equation and solve for x:
y = -2x + 12
Substituting the value of y:
(20 - x)/4 = -2x + 12
Next, let's cross-multiply to eliminate the fraction:
20 - x = -8x + 48
Rearrange the equation to isolate x:
7x = 28
Divide by 7:
x = 4
Now that we have the value of x, we can substitute it back into one of the original equations to solve for y. Let's use the first equation:
y = -2(4) + 12
y = 4
So, we have found a unique solution to the system of equations, which means there is only one solution.
To determine the number of solutions in a system of equations, you can use different methods such as graphing, substitution, or elimination. Let's use substitution to solve this system of equations:
y = -2x + 12 [Equation 1]
4y + x - 20 = 0 [Equation 2]
Step 1: Solve Equation 1 for y:
y = -2x + 12
Step 2: Substitute this value of y into Equation 2:
4(-2x + 12) + x - 20 = 0
Step 3: Simplify and solve for x:
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = 4
Step 4: Substitute the value of x back into Equation 1 to find y:
y = -2(4) + 12
y = -8 + 12
y = 4
So, the system of equations has one solution, which is x = 4 and y = 4.