To solve this system of linear equations using the method of elimination, follow these steps:
We have the two equations:
1) \(\( 2x + y = 12 \)\)
2) \(\( -3x + y = 2 \)\)
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 \(\( y \)\) in this case.
To eliminate \(\( y \)\), we need to make the coefficient of \(\( y \)\) 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) \(\( 2x + y = 12 \)\) (No change here)
2) \(\( -1(-3x + y) = -1(2) \)\) (Multiply the entire equation by -1)
After multiplying, we get the new form of the second equation:
2) \(\( 3x - y = -2 \)\)
Now, add equations 1) and 2) together to eliminate \(\( y \)\):
\(\( (2x + y) + (3x - y) = 12 + (-2) \)\)
You will notice that \(\( y \)\) cancels out:
\(\( 5x = 10 \)\)
Now, divide both sides by 5 to solve for \(\( x \)\):
\(\( x = \frac{10}{5} \)\)
\(\( x = 2 \)\)
Now that we have the value of \(\( x \)\), we can substitute it into one of the original equations to solve for \(\( y \)\). Let's use the first equation:
\(\( 2x + y = 12 \)\)
Substitute \(\( x = 2 \)\) into the equation:
\(\( 2(2) + y = 12 \)\)
\(\( 4 + y = 12 \)\)
Now, subtract 4 from both sides to solve for \(\( y \)\):
\(\( y = 12 - 4 \)\)
\(\( y = 8 \)\)
Therefore, the solution to the system of equations is \(\( x = 2 \)\) and \(\( y = 8 \)\).
