To solve the given system of equations:
[tex]\[
\begin{cases}
5x + 2y = 1 \\
-3x + 3y = 5
\end{cases}
\][/tex]
we will use the method of solving simultaneous equations.
### Step 1: Isolate one of the variables in one of the equations
We start by isolating [tex]\(y\)[/tex] in the second equation. The second equation is:
[tex]\[
-3x + 3y = 5
\][/tex]
Divide the entire equation by 3 to simplify:
[tex]\[
-x + y = \frac{5}{3}
\][/tex]
Rewriting it, we get:
[tex]\[
y = x + \frac{5}{3}
\][/tex]
### Step 2: Substitute the expression for [tex]\(y\)[/tex] into the first equation
Next, we substitute [tex]\(y = x + \frac{5}{3}\)[/tex] into the first equation:
[tex]\[
5x + 2y = 1
\][/tex]
Substituting [tex]\(y\)[/tex]:
[tex]\[
5x + 2\left(x + \frac{5}{3}\right) = 1
\][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, we solve for [tex]\(x\)[/tex]:
[tex]\[
5x + 2x + \frac{10}{3} = 1
\][/tex]
Combine like terms:
[tex]\[
7x + \frac{10}{3} = 1
\][/tex]
Next, subtract [tex]\(\frac{10}{3}\)[/tex] from both sides:
[tex]\[
7x = 1 - \frac{10}{3}
\][/tex]
Convert 1 to a fraction with a denominator of 3:
[tex]\[
7x = \frac{3}{3} - \frac{10}{3}
\][/tex]
Simplify the right-hand side:
[tex]\[
7x = \frac{3 - 10}{3}
\][/tex]
[tex]\[
7x = -\frac{7}{3}
\][/tex]
Now, divide both sides by 7:
[tex]\[
x = -\frac{7}{3} \div 7
\][/tex]
[tex]\[
x = -\frac{7}{3} \times \frac{1}{7}
\][/tex]
[tex]\[
x = -\frac{1}{3}
\][/tex]
### Step 4: Substitute [tex]\(x\)[/tex] back into the expression for [tex]\(y\)[/tex]
Now that we have [tex]\(x = -\frac{1}{3}\)[/tex], we substitute this value back into the expression for [tex]\(y\)[/tex]:
[tex]\[
y = x + \frac{5}{3}
\][/tex]
[tex]\[
y = -\frac{1}{3} + \frac{5}{3}
\][/tex]
Simplify the fraction:
[tex]\[
y = \frac{-1 + 5}{3}
\][/tex]
[tex]\[
y = \frac{4}{3}
\][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[
x = -\frac{1}{3}, \quad y = \frac{4}{3}
\][/tex]
So the solution is:
[tex]\[
\left( x, y \right) = \left( -\frac{1}{3}, \frac{4}{3} \right)
\][/tex]
