We are given a system of two linear equations with two variables, $x$ and $y$. The system is: $5x + 2y = 1$ $-3x + 3y = 5$ We are asked to solve for $x$ and $y$.

1. Problem Description

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

$5x + 2y = 1$

$-3x + 3y = 5$

We are asked to solve for $x$ and $y$.

2. Solution Steps

We will use the method of elimination to solve for $x$ and $y$.

First, multiply the first equation by $3$ and the second equation by $-2$ to eliminate the $y$ variable.

$3(5x + 2y) = 3(1)$

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

This gives us:

$15x + 6y = 3$

$6x - 6y = -10$

Now, add the two equations together:

$(15x + 6y) + (6x - 6y) = 3 + (-10)$

$21x = -7$

$x = -\frac{7}{21} = -\frac{1}{3}$

Now that we have the value of $x$, we can substitute it into one of the original equations to solve for $y$. Let's use the first equation:

$5x + 2y = 1$

$5(-\frac{1}{3}) + 2y = 1$

$-\frac{5}{3} + 2y = 1$

$2y = 1 + \frac{5}{3}$

$2y = \frac{3}{3} + \frac{5}{3}$

$2y = \frac{8}{3}$

$y = \frac{8}{3} \cdot \frac{1}{2}$

$y = \frac{4}{3}$

Thus, the solution is $x = -\frac{1}{3}$ and $y = \frac{4}{3}$.

3. Final Answer

$x = -\frac{1}{3}$

$y = \frac{4}{3}$