Answer
$x=3262,y=2015,z=1796$
Work Step by Step
We denote that the spending on housing by $x$, the spending on Vehicle/gas by $y$ and the spending on health care by $z$.
We are given the system:
$\begin{cases}
x+y+z=7073\\
x-y=1247\\
x-z=1466
\end{cases}$
We will use the addition method. Add Equation 2 to Equation 3. Add a new combined Equation to Equation 1 to eliminate $y$ and $z$:
$\begin{cases}
x-y+(x-z)=1247+(1466),\\
x+y+z+(2x-y-z)=7073+(2713)
\end{cases}$
$\begin{cases}
3x=9786, x=3262\\
\end{cases}$
substituting the result we got for $x$ in to the original second equation, we can get an answer for $y$.
$\begin{cases}
3262-y=1247, y=2015\\
\end{cases}$
substituing the result we got for $x$ in to the original third equation we can get an answer for $z$.
$\begin{cases}
3262-x=1466, z=1796\\
\end{cases}$
$x=3262,y=2015,z=1796$
