Answer
$(x,y)= (2,15)$
Work Step by Step
Given: $$\begin{cases}
y &= 4x+7\\
x+3y &=47.
\end{cases}$$ First, rewrite the second equation in slope intercept form.
$$\begin{aligned}
x+3y &=47 \\
3y &=-x+47 \\
y &= -\frac{1}{3}x+\frac{47}{3}.
\end{aligned}$$ Compare the two equations: $$\begin{cases}
y &= 4x+7\\
y &=-\frac{1}{3}x+\frac{47}{3}.
\end{cases}$$ By comparison, we see that both lines are not the same. This means that the lines are independent and have one solution. Lets solve the original system by substituting for $y$ from the first equation into the second.
$$\begin{aligned}
x+3\left(4x+7\right) &= 47\\
x+12x+21&= 47\\
13x&=47-21 \\
13x&= 26\\
x&=\frac{26}{13}\\
&= 2.
\end{aligned}$$ Find the value of $y$ using either of the equations. $$\begin{aligned}
y &= 4\cdot 2+7\\
&=15\\
y&=-\frac{1}{3}\cdot 2+\frac{47}{3} \\
&= \frac{47-2}{3}\\
&= 15.
\end{aligned}$$ The solution set is $(x,y)= (2,15)$ . Warrick's solution is correct and the one given by Shania is incomplete. She correctly found the value of $x$ but failed to compute the value of $y$. She also failed to say anything about the nature of the system, whether the solution is consistent, inconsistent or dependent.
