To solve the given system of equations:
[tex]\[
\begin{array}{l}
2y = -x + 9 \\
3x - 6y = -15
\end{array}
\][/tex]
we can follow these steps:
1. Re-arrange the first equation to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
2y = -x + 9
\][/tex]
Divide both sides by 2:
[tex]\[
y = \frac{-x + 9}{2}
\][/tex]
2. Substitute the expression for [tex]\( y \)[/tex] into the second equation:
The second equation is:
[tex]\[
3x - 6y = -15
\][/tex]
Substitute [tex]\( y \)[/tex] from step 1:
[tex]\[
3x - 6 \left( \frac{-x + 9}{2} \right) = -15
\][/tex]
3. Simplify and solve for [tex]\( x \)[/tex]:
Distribute the -6 inside the parentheses:
[tex]\[
3x - 6 \left( \frac{-x}{2} + \frac{9}{2} \right) = -15
\][/tex]
[tex]\[
3x - 6 \left( -\frac{x}{2} + \frac{9}{2} \right) = -15
\][/tex]
[tex]\[
3x - \left( -3x + 27 \right) = -15
\][/tex]
[tex]\[
3x + 3x - 27 = -15
\][/tex]
Combine like terms:
[tex]\[
6x - 27 = -15
\][/tex]
Add 27 to both sides:
[tex]\[
6x = 12
\][/tex]
Divide by 6:
[tex]\[
x = 2
\][/tex]
4. Substitute [tex]\( x = 2 \)[/tex] back into the expression for [tex]\( y \)[/tex]:
From step 1, we have:
[tex]\[
y = \frac{-x + 9}{2}
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
y = \frac{-2 + 9}{2}
\][/tex]
[tex]\[
y = \frac{7}{2}
\][/tex]
Therefore, the solution to the system is:
[tex]\[
\left( 2, \frac{7}{2} \right)
\][/tex]
So in the correct answer box for the solution to the system, we have:
The solution to the system is [tex]\( \left( 2, \frac{7}{2} \right) \)[/tex].
