System Equations Setup

We are given two equations:

1. $ 2x - 3y = -1 $
2. $ \frac{x}{x+y} = \frac{7}{12} $

Our goal is to find the value of $2xy$. First, let's simplify the second equation.

Solving Variables x and y

From the second equation, $\frac{x}{x+y} = \frac{7}{12}$, we cross-multiply:

$ 12x = 7(x+y) $ $ 12x = 7x + 7y $

Rearrange to find a relationship between $x$ and $y$:

$ 12x - 7x = 7y $ $ 5x = 7y $

We can express $x$ in terms of $y$: $ x = \frac{7}{5}y $

Now, substitute this expression for $x$ into the first equation, $2x - 3y = -1$:

$ 2\left(\frac{7}{5}y\right) - 3y = -1 $ $ \frac{14}{5}y - 3y = -1 $

To solve for $y$, find a common denominator:

$ \frac{14}{5}y - \frac{15}{5}y = -1 $ $ \frac{14 - 15}{5}y = -1 $ $ -\frac{1}{5}y = -1 $

Multiply both sides by -5:

$ y = 5 $

Substitute the value $y=5$ back into the relation $x = \frac{7}{5}y$:

$ x = \frac{7}{5}(5) $ $ x = 7 $

So, the values are $x=7$ and $y=5$.

Calculating $2xy$ Value

Finally, calculate the value of $2xy$ using $x=7$ and $y=5$:

$ 2xy = 2(7)(5) $ $ 2xy = 2(35) $ $ 2xy = 70 $