Solve using the elimination method. Show your work. If the system has not solution or an infinite number of solutions, state this.
-2x-8y=-42
-4x+6y=-18
Multiply top equation by 2 and then subtract equation #2 from #1.
To solve the system of equations using the elimination method, we need to get rid of one variable by multiplying one or both equations by suitable constants in order to make the coefficients of that variable the same in both equations. Then, we can add or subtract the equations to eliminate that variable.
Let's start by getting rid of the variable x. We will multiply the second equation by -2 to make the coefficients of x in both equations the same:
-2 * (-4x+6y) = -2 * (-18)
8x - 12y = 36
Now we have the system of equations:
-2x - 8y = -42
8x - 12y = 36
Next, we will subtract the first equation from the second equation to eliminate x:
(8x - 12y) - (-2x - 8y) = 36 - (-42)
8x + 2x - 12y + 8y = 36 + 42
10x - 4y = 78
Now we have the system of equations:
-2x - 8y = -42
10x - 4y = 78
To eliminate y, we will multiply the first equation by -1/4:
(-1/4) * (-2x - 8y) = (-1/4) * (-42)
(1/2)x + 2y = 21
Now we have the system of equations:
(1/2)x + 2y = 21
10x - 4y = 78
Now, we can add the equations to eliminate y:
(1/2)x + 2y + 10x - 4y = 21 + 78
(1/2 + 10)x + (2 - 4)y = 99
(21/2)x - 2y = 99
The system of equations now becomes:
(21/2)x - 2y = 99
10x - 4y = 78
Now we can see that we have two equations with the same variables. This means we can't eliminate any more variables.
To check if this system has a solution, we can compare the coefficients. Notice that if we multiply the first equation by 5/21, we get:
(5/21) * (21/2)x - (5/21) * 2y = (5/21) * 99
x - (10/21)y = 15
This equation is different from the second equation. Therefore, the system of equations has no solution.
To summarize, the system of equations -2x - 8y = -42 and -4x + 6y = -18 has no solution.