Sure, let's solve the given system of equations step-by-step using substitution:
[tex]\[
\begin{aligned}
(1) & \quad x - 3y = 2 \\
(2) & \quad 2x + 5y = 15
\end{aligned}
\][/tex]
### Step 1: Solve one of the equations for one variable
Let's solve equation (1) for [tex]\( x \)[/tex]:
[tex]\[
x - 3y = 2 \quad \Rightarrow \quad x = 2 + 3y
\][/tex]
### Step 2: Substitute this expression into the other equation
Now we'll substitute [tex]\( x = 2 + 3y \)[/tex] into equation (2):
[tex]\[
2(2 + 3y) + 5y = 15
\][/tex]
### Step 3: Simplify and solve for [tex]\( y \)[/tex]
First, distribute the 2 on the left side:
[tex]\[
4 + 6y + 5y = 15
\][/tex]
Combine like terms:
[tex]\[
4 + 11y = 15
\][/tex]
Subtract 4 from both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[
11y = 11
\][/tex]
Divide both sides by 11:
[tex]\[
y = 1
\][/tex]
### Step 4: Substitute [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]
We have found [tex]\( y = 1 \)[/tex]. Substitute this value back into the expression [tex]\( x = 2 + 3y \)[/tex]:
[tex]\[
x = 2 + 3(1) \quad \Rightarrow \quad x = 2 + 3 \quad \Rightarrow \quad x = 5
\][/tex]
### Step 5: Write the solution
The solution to the system of equations is:
[tex]\[
\boxed{(x, y) = (5, 1)}
\][/tex]
