To solve this system of linear equations using the method of elimination, follow these steps:
We have the two equations:
1) [tex]\( 2x + y = 12 \)[/tex]
2) [tex]\( -3x + y = 2 \)[/tex]
The goal of the elimination method is to add or subtract the equations to eliminate one variable, making it possible to solve for the other variable. We choose to eliminate [tex]\( y \)[/tex] in this case.
To eliminate [tex]\( y \)[/tex], we need to make the coefficient of [tex]\( y \)[/tex] in both equations equal with opposite signs. We see that they are already the same, but with the same sign, so by multiplying the second equation by -1, we can obtain an opposite sign:
1) [tex]\( 2x + y = 12 \)[/tex] (No change here)
2) [tex]\( -1(-3x + y) = -1(2) \)[/tex] (Multiply the entire equation by -1)
After multiplying, we get the new form of the second equation:
2) [tex]\( 3x - y = -2 \)[/tex]
Now, add equations 1) and 2) together to eliminate [tex]\( y \)[/tex]:
[tex]\( (2x + y) + (3x - y) = 12 + (-2) \)[/tex]
You will notice that [tex]\( y \)[/tex] cancels out:
[tex]\( 5x = 10 \)[/tex]
Now, divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\( x = \frac{10}{5} \)[/tex]
[tex]\( x = 2 \)[/tex]
Now that we have the value of [tex]\( x \)[/tex], we can substitute it into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use the first equation:
[tex]\( 2x + y = 12 \)[/tex]
Substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\( 2(2) + y = 12 \)[/tex]
[tex]\( 4 + y = 12 \)[/tex]
Now, subtract 4 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\( y = 12 - 4 \)[/tex]
[tex]\( y = 8 \)[/tex]
Therefore, the solution to the system of equations is [tex]\( x = 2 \)[/tex] and [tex]\( y = 8 \)[/tex].
