To solve the system of equations:
[tex]\[
\left\{
\begin{aligned}
3x - 2y &= 4 \\
y &= 3x - 5
\end{aligned}
\right.
\][/tex]
We can use the substitution method, where we substitute one equation into the other.
Step-by-step solution:
1. From the second equation [tex]\( y = 3x - 5 \)[/tex], we can use this expression for [tex]\( y \)[/tex] in the first equation.
[tex]\[
3x - 2(3x - 5) = 4
\][/tex]
2. Distribute the [tex]\(-2\)[/tex] through the parenthesis:
[tex]\[
3x - 6x + 10 = 4
\][/tex]
3. Combine like terms:
[tex]\[
-3x + 10 = 4
\][/tex]
4. Subtract 10 from both sides of the equation to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-3x = 4 - 10
\][/tex]
[tex]\[
-3x = -6
\][/tex]
5. Divide both sides by [tex]\(-3\)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-6}{-3} = 2
\][/tex]
6. Now that we have [tex]\( x = 2 \)[/tex], we substitute this value back into the second original equation to find [tex]\( y \)[/tex]:
[tex]\[
y = 3x - 5
\][/tex]
[tex]\[
y = 3(2) - 5
\][/tex]
[tex]\[
y = 6 - 5
\][/tex]
[tex]\[
y = 1
\][/tex]
So, the solution to the system of equations is:
[tex]\[
\boxed{x = 2, y = 1}
\][/tex]
