To solve this system of equations using the elimination method, we need to eliminate one of the variables. One way to do this is to add the two equations together in a way that will eliminate one of the variables.
First, we can multiply the second equation by -1 to change the sign of all its terms:
-6x + 5y = 1
-6x - 4y = 10
Now we can add the two equations together to eliminate x:
( -6x + 5y ) + ( -6x - 4y ) = 1 + 10
Simplifying the left side and the right side:
-12x + y = 11
Now we have one equation with one variable, y. We can solve for y by isolating it on one side of the equation:
-12x + y = 11
y = 12x + 11
We can substitute this expression for y into either of the original equations to solve for x. Let's use the first equation:
-6x + 5y = 1
-6x + 5(12x + 11) = 1
Simplifying the left side:
54x + 55 = 1
Subtracting 55 from both sides:
54x = -54
Dividing both sides by 54:
x = -1
So the solution to the system of equations is x = -1 and y = 1. We can check this solution by substituting these values into both original equations and verifying that they are true.