Answer
$(g,f)= (-14.5,-89.5)$ (consistent independent system)
Work Step by Step
Given: $$\begin{cases}
f &= 7(g+2)-2\\
&= 7g+12\\
f& =5(g-6)+13\\
&= 5g-17
\end{cases}$$ We are asked to use the method of substitution to solve the given system. Do the substitution and solve: $$\begin{aligned}
7g+12 &= 5g-17\\
7g-5g &=-17-12 \\
2g &= -29\\
g&= -\frac{29}{2}\\
&= -14.5.
\end{aligned}$$ Now find $f$ using any one of the equations given.
$$\begin{aligned}
f &= 5\cdot(-14.5)-17\\
&=-89.5.
\end{aligned}$$ Check $$\begin{aligned}
-89.5 &\stackrel{?}{=}7\cdot (-14.5) +12\\
-89.5& = -89.5\checkmark\\
-89.5 &\stackrel{?}{=}5\cdot (-14.5) -17\\
-89.5& = -89.5\checkmark.
\end{aligned}$$ The solution set is $(g,f)= (-14.5,-89.5)$ . The system is consistent and independent because it has a single solution.
