How many solutions does the system of equations have?
y=−2x+12
4y+x−20=0
To determine the number of solutions, we need to find the intersection point of the two equations.
First, let's solve the system of equations by substitution.
Substitute the value of y from the first equation into the second equation:
4(-2x + 12) + x - 20 = 0
Simplify the equation:
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = -28 / -7
x = 4
Now substitute the value of x back into the first equation to find the value of y:
y = -2(4) + 12
y = -8 + 12
y = 4
Therefore, the system of equations has one solution at (4, 4).
The system of equations has exactly one solution.
To determine the number of solutions for the given system of equations, we need to solve the equations and then assess the relationship between the variables.
First, let's rearrange the second equation to solve for x:
4y + x - 20 = 0
x = 20 - 4y
Now substitute this expression for x into the first equation:
y = -2(20 - 4y) + 12
y = -40 + 8y + 12
Combine like terms:
-7y = -28
Divide both sides by -7:
y = 4
Substitute this value for y back into either of the original equations (let's use the first one):
y = -2x + 12
4 = -2x + 12
-2x = 4 - 12
-2x = -8
Divide both sides by -2:
x = 4
Therefore, the system of equations has a unique solution, which is (x, y) = (4, 4).
To determine the number of solutions that the system of equations has, we can solve the equations simultaneously and check the relationship between the variables. Here's how you can do it:
1. Begin by rewriting the equations in standard form:
Equation 1: y = -2x + 12
Equation 2: x + 4y = 20 (Rearrange the equation to put it in standard form)
2. Choose a method to solve the system. In this case, let's use the substitution method:
i. Solve Equation 1 for y in terms of x:
y = -2x + 12
ii. Substitute this expression for y in Equation 2:
x + 4(-2x + 12) = 20
3. Simplify and solve for x:
x - 8x + 48 = 20
-7x + 48 = 20
-7x = 20 - 48
-7x = -28
x = -28 / -7
x = 4
4. Substitute the value of x back into either equation to find the value of y:
y = -2(4) + 12
y = -8 + 12
y = 4
5. Check whether the values of x and y satisfy the second equation:
4y + x - 20 = 0
4(4) + 4 - 20 = 0
16 + 4 - 20 = 0
20 - 20 = 0
0 = 0
6. Analyze the results:
The system of equations has a unique solution, which means there is only one solution that satisfies both equations. In this case, the solution is x = 4 and y = 4.
Therefore, the system of equations has exactly one solution.