Answer
The solution to this system of equations is $f = 4$ and $g = -8$.
Work Step by Step
When we use elimination to solve a system of equations, we need to make sure that one of the variables in both equations is the same but differing only in sign.
Let's take a look at the system of equations in this exercise. In this case, we already have one variable term in both equations that is the same value but with opposite signs:
$7f + g = 20$
$3f - g = 20$
Let's add the two equations:
$10f = 40$
Divide each side by $10$ to solve for $f$:
$f = 4$
Substitute this value for $f$ into one of the equations to solve for $g$:
$3(4) - g = 20$
Multiply first:
$12 - g = 20$
Collect constant terms on the right side of the equation:
$-g = 8$
Divide each side by $-1$ to solve for $g$:
$g = -8$
The solution to this system of equations is $f = 4$ and $g = -8$.
To see if this solution is correct, substitute in the values we just found for $g$ and $h$ into one of the original equations:
$3(4) - (-8) = 20$
Multiply first:
$12 + 8 = 20$
Add:
$20 = 20$
The two sides are equal; therefore, this solution is correct.
This system of equations is consistent because it has at least one solution. It is also independent because there is only one solution.