Solve the following system of linear equations by addition. Indicate whether the given system of linear equations has one solution, has no solution, or has an infinite number of solutions. If the system has one solution, find the solution.
{−8x+2y=14 −4x+y=4
To use the method of addition, we need to eliminate one variable by adding the two equations together.
First, we can start by eliminating the y variable. To do this, we can multiply the second equation by 2 and add the two equations together:
-8x + 2y + (-8x + 2y) = 14 + (2 * 4)
-16x + 4y = 14 + 8
-16x + 4y = 22
Now we have a new equation: -16x + 4y = 22.
We can subtract this equation from the original first equation (-8x + 2y = 14):
(-8x + 2y) - (-16x + 4y) = 14 - 22
-8x + 2y + 16x - 4y = -8
8x - 2y = -8
Now we have a new equation: 8x - 2y = -8.
Next, we can multiply the second equation by 8 and subtract the new equation from the original second equation (-4x + y = 4):
(-4x + y) - (8x - 2y) = 4 - (-8)
-4x + y - 8x + 2y = 4 + 8
-12x + 3y = 12
Now we have a new equation: -12x + 3y = 12.
We can simplify this system of linear equations:
1) -16x + 4y = 22
2) 8x - 2y = -8
3) -12x + 3y = 12
Now we can solve the system of equations using elimination or substitution. Let's use elimination in this case.
To eliminate the variable "x" from equation 1 and equation 2, we can multiply equation 2 by 2:
(2) * (8x - 2y) = (2) * (-8)
16x - 4y = -16
Now we have a new equation: 16x - 4y = -16.
We can add this equation to equation 1:
(-16x + 4y) + (16x - 4y) = 22 - 16
0 = 6
This equation simplifies to 0 = 6, which is not true.
Therefore, the system of equations has no solution.