We are given a system of two linear equations with two variables $x$ and $y$: $(m+1)x - my = 4$ $3x - 5y = m$ We are also given that $m = 2$. We need to solve this system for $x$ and $y$.

1. Problem Description

We are given a system of two linear equations with two variables $x$ and $y$:

$(m+1)x - my = 4$

$3x - 5y = m$

We are also given that $m = 2$. We need to solve this system for $x$ and $y$.

2. Solution Steps

First, substitute $m = 2$ into the equations:

$(2+1)x - 2y = 4$

$3x - 5y = 2$

This simplifies to:

$3x - 2y = 4$

$3x - 5y = 2$

Now we have the system:

$3x - 2y = 4$ (1)

$3x - 5y = 2$ (2)

We can subtract equation (2) from equation (1) to eliminate $x$:

$(3x - 2y) - (3x - 5y) = 4 - 2$

$3x - 2y - 3x + 5y = 2$

$3y = 2$

$y = \frac{2}{3}$

Now, substitute $y = \frac{2}{3}$ into equation (1):

$3x - 2(\frac{2}{3}) = 4$

$3x - \frac{4}{3} = 4$

$3x = 4 + \frac{4}{3}$

$3x = \frac{12}{3} + \frac{4}{3}$

$3x = \frac{16}{3}$

$x = \frac{16}{3} \cdot \frac{1}{3}$

$x = \frac{16}{9}$

3. Final Answer

$x = \frac{16}{9}$

$y = \frac{2}{3}$