Let's solve the system of equations step-by-step to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
The given system of equations is:
[tex]\[
\begin{array}{l}
5x + 2y = 9 \\
2x - 3y = 15
\end{array}
\][/tex]
### Step 1: Solve one of the equations for one variable
First, we can solve the first equation for [tex]\( y \)[/tex]:
[tex]\[
5x + 2y = 9 \quad \Rightarrow \quad 2y = 9 - 5x \quad \Rightarrow \quad y = \frac{9 - 5x}{2}
\][/tex]
### Step 2: Substitute the expression for [tex]\( y \)[/tex] into the second equation
Now, substitute [tex]\( y = \frac{9 - 5x}{2} \)[/tex] into the second equation:
[tex]\[
2x - 3 \left( \frac{9 - 5x}{2} \right) = 15
\][/tex]
### Step 3: Simplify the equation
Distribute the [tex]\(-3\)[/tex]:
[tex]\[
2x - \frac{27 - 15x}{2} = 15
\][/tex]
Multiply through by 2 to clear the fraction:
[tex]\[
4x - (27 - 15x) = 30
\][/tex]
Simplify inside the parentheses:
[tex]\[
4x - 27 + 15x = 30
\][/tex]
Combine like terms:
[tex]\[
19x - 27 = 30
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Add 27 to both sides:
[tex]\[
19x = 57
\][/tex]
Divide by 19:
[tex]\[
x = 3
\][/tex]
### Step 5: Solve for [tex]\( y \)[/tex]
Now, substitute [tex]\( x = 3 \)[/tex] back into the expression we found for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{9 - 5(3)}{2} = \frac{9 - 15}{2} = \frac{-6}{2} = -3
\][/tex]
### Result
So, the solution to the system of equations is:
[tex]\[
x = 3 \quad \text{and} \quad y = -3
\][/tex]
Thus, the correct answer is
[tex]\[ \boxed{(3, -3)} \][/tex]
So, the answer is:
D. [tex]\((3, -3)\)[/tex]
