To solve the given system of equations, we can use the method of substitution. Here is a step-by-step solution:
Given system of equations:
[tex]\[
\begin{array}{l}
2y = -x + 9 \tag{1} \\
3x - 6y = -15 \tag{2}
\end{array}
\][/tex]
Step 1: Solve equation (1) for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex].
[tex]\[
2y = -x + 9
\][/tex]
Divide both sides by 2 to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-x + 9}{2}
\][/tex]
Step 2: Substitute the expression for [tex]\(y\)[/tex] from Step 1 into equation (2).
[tex]\[
3x - 6 \left(\frac{-x + 9}{2}\right) = -15
\][/tex]
Step 3: Simplify the substituted equation.
[tex]\[
3x - 3(-x + 9) = -15
\][/tex]
Distribute the [tex]\(-3\)[/tex]:
[tex]\[
3x + 3x - 27 = -15
\][/tex]
Combine like terms:
[tex]\[
6x - 27 = -15
\][/tex]
Step 4: Solve for [tex]\(x\)[/tex].
[tex]\[
6x = -15 + 27
\][/tex]
[tex]\[
6x = 12
\][/tex]
[tex]\[
x = \frac{12}{6}
\][/tex]
[tex]\[
x = 2
\][/tex]
Step 5: Substitute [tex]\(x = 2\)[/tex] back into the expression for [tex]\(y\)[/tex] from Step 1.
[tex]\[
y = \frac{-x + 9}{2}
\][/tex]
[tex]\[
y = \frac{-2 + 9}{2}
\][/tex]
[tex]\[
y = \frac{7}{2}
\][/tex]
[tex]\[
y = 3.5
\][/tex]
So, the solution to the system of equations is [tex]\((x, y) = (2, 3.5)\)[/tex].
Therefore, the solution to the system is [tex]\(\boxed{2}, \boxed{3.5}\)[/tex].
