Solve by the linear combination method (with or without multiplication).
6x – 2y = 2
–3x + 4y = 5
easiest way:
double the 2nd, then add that to the first
6y = 12
y = 2
sub back into the 1st
6x - 4 = 2
6x = 6
x = 1
To solve this system of equations using the linear combination method, we need to eliminate one variable by adding or subtracting the two equations. The goal is to create a new equation where one variable is eliminated so that we can solve for the remaining variable.
Let's start by eliminating the x-variable. We can do this by multiplying the first equation by 3 and the second equation by 2. This will allow us to cancel out the x-variable when we add the two equations together.
First equation: 6x – 2y = 2
Multiply by 3: 18x - 6y = 6
Second equation: -3x + 4y = 5
Multiply by 2: -6x + 8y = 10
Now, we can add the equations together to eliminate the x-variable:
(18x - 6y) + (-6x + 8y) = 6 + 10
12x + 2y = 16
Now, we have a new equation with only the y-variable, which we can solve for. Rearranging the equation to solve for y:
2y = -12x + 16
y = (-12x + 16) / 2
y = -6x + 8
Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the first equation:
6x - 2y = 2
6x - 2(-6x + 8) = 2
6x + 12x - 16 = 2
18x - 16 = 2
18x = 18
x = 1
The solution to the system of equations is x = 1 and y = -6x + 8.