Sure! Let's solve the given system of equations step-by-step.
We have the following system of linear equations:
[tex]\[
\begin{cases}
5x - 2y = 23 \\
2x - 3y = 18
\end{cases}
\][/tex]
Step 1: Express one of the variables in terms of the other using the first equation.
From the first equation \(5x - 2y = 23\), we can express \(y\) in terms of \(x\):
[tex]\[
5x - 2y = 23 \quad \Rightarrow \quad 2y = 5x - 23 \quad \Rightarrow \quad y = \frac{5x - 23}{2}
\][/tex]
Step 2: Substitute this expression for \(y\) into the second equation.
Substitute \(y = \frac{5x - 23}{2}\) into the second equation \(2x - 3y = 18\):
[tex]\[
2x - 3\left(\frac{5x - 23}{2}\right) = 18
\][/tex]
Step 3: Solve for \(x\).
Let's clear the fraction by multiplying everything by 2:
[tex]\[
2 \cdot 2x - 3 \cdot (5x - 23) = 18 \cdot 2 \quad \Rightarrow \quad 4x - 3(5x - 23) = 36
\][/tex]
Distribute the -3:
[tex]\[
4x - 15x + 69 = 36 \quad \Rightarrow \quad -11x + 69 = 36
\][/tex]
Solve for \(x\):
[tex]\[
-11x = 36 - 69 \quad \Rightarrow \quad -11x = -33 \quad \Rightarrow \quad x = \frac{-33}{-11} \quad \Rightarrow \quad x = 3
\][/tex]
Step 4: Use the value of \(x\) to find \(y\).
Substitute \(x = 3\) back into \(y = \frac{5x - 23}{2}\):
[tex]\[
y = \frac{5(3) - 23}{2} = \frac{15 - 23}{2} = \frac{-8}{2} = -4
\][/tex]
Therefore, the solution to the system is:
[tex]\[
x = 3, \quad y = -4
\][/tex]
