Sure, let's solve the system of equations step by step:
[tex]\[
\begin{cases}
3x + 4y = 3 \\
2x - y = 13
\end{cases}
\][/tex]
Step 1: Solve one of the equations for one of the variables
We can start by solving the second equation for [tex]\( y \)[/tex]:
[tex]\[
2x - y = 13
\][/tex]
First, isolate [tex]\( y \)[/tex]:
[tex]\[
-y = 13 - 2x \implies y = 2x - 13
\][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] into the first equation
Now, we substitute [tex]\( y = 2x - 13 \)[/tex] into the first equation:
[tex]\[
3x + 4(2x - 13) = 3
\][/tex]
Step 3: Simplify and solve for [tex]\( x \)[/tex]
Distribute the 4 inside the parentheses:
[tex]\[
3x + 8x - 52 = 3
\][/tex]
Combine like terms:
[tex]\[
11x - 52 = 3
\][/tex]
Add 52 to both sides:
[tex]\[
11x = 55
\][/tex]
Divide both sides by 11:
[tex]\[
x = 5
\][/tex]
Step 4: Substitute the value of [tex]\( x \)[/tex] back into the equation for [tex]\( y \)[/tex]
We know that [tex]\( y = 2x - 13 \)[/tex]. Substitute [tex]\( x = 5 \)[/tex] into this equation:
[tex]\[
y = 2(5) - 13
\][/tex]
Simplify:
[tex]\[
y = 10 - 13
\][/tex]
[tex]\[
y = -3
\][/tex]
Conclusion:
The solution to the system of equations is [tex]\( x = 5 \)[/tex] and [tex]\( y = -3 \)[/tex].
So the solution is:
[tex]\[
(x, y) = \boxed{(5, -3)}
\][/tex]
