#### Answer

The wine company should mix $100\text{ gallons}$ of $5\%$ California wine with $\text{100 gallons}$ of $9\%$ French wine.

#### Work Step by Step

Let $x$ represent the gallons of $5\text{ percent}$ California wine and $y$ represent the gallons of $9\text{ percent}$ French wine.
The amount of pure wine in each solution is found by multiplying the amount of gallons by the concentration rate.
There are two unknown quantities; therefore, a system of two independent equations with $x$ and $y$ is set up.
Amount of wine 5 percent mixture + Amount of wine 9 percent mixture=Amount of wine 7 percent mixture
And:
Amount of pure wine in 5 percent mixture + Amount of pure wine in 9 percent mixture = Amount of pure wine in 7 percent mixture
Consider the equation,
$\begin{align}
& x+y=200 \\
& x=200-y
\end{align}$ …… (1)
And
$0.05x+0.09y=14$ …… (2)
Substitute $200-y$for $x$ in equation $\left( 2 \right)$ to get,
$\begin{align}
& 0.05\left( 200-y \right)+0.09y=14 \\
& 10-0.05y+0.09y=14 \\
& 0.04y=4
\end{align}$
Divide the above equation by $0.04$ to get,
$\begin{align}
& \frac{0.04y}{0.04}=\frac{4}{0.04} \\
& y=100
\end{align}$
Substitute $100$ for $y$ in the equation $\left( 1 \right)$ to get,
$\begin{align}
& x=200-100 \\
& x=100 \\
\end{align}$
Check: $\left( 100,100 \right)$
Put $x=100$ and $y=100$ in the equation (1),
$\begin{align}
x+y=200 & \\
100+100\overset{?}{\mathop{=}}\,200 & \\
200=200 & \\
\end{align}$
And put $x=100$and $y=100$ in the equation (2),
$\begin{align}
0.05\left( 100 \right)+0.09\left( 100 \right)\overset{?}{\mathop{=}}\,14 & \\
5+9\overset{?}{\mathop{=}}\,14 & \\
14=14 & \\
\end{align}$
The ordered pair $\left( 100,100 \right)$ satisfies both equations.
Hence, the wine company should mix $100\text{ gallons}$ of $5\%$ California wine with $\text{100 gallons}$ of $9\%$ French wine.