Certainly! Let’s solve the given system of equations step-by-step using the substitution method.
The system of equations is:
[tex]\[
\begin{array}{l}
2y = -x + 9 \\
3x - 6y = -15
\end{array}
\][/tex]
Step 1: Solve the first equation for [tex]\( y \)[/tex]
Starting with the first equation:
[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] into the second equation
Now, take the expression we found for [tex]\( y \)[/tex] and substitute it into the second equation:
[tex]\[
3x - 6\left(\frac{-x + 9}{2}\right) = -15
\][/tex]
Distribute [tex]\(-6\)[/tex] inside the parentheses:
[tex]\[
3x - 3(-x + 9) = -15
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
3x + 3x - 27 = -15
\][/tex]
Combine like terms:
[tex]\[
6x - 27 = -15
\][/tex]
Step 3: Solve for [tex]\( x \)[/tex]
Add 27 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
6x - 27 + 27 = -15 + 27
\][/tex]
Simplify both sides:
[tex]\[
6x = 12
\][/tex]
Divide by 6:
[tex]\[
x = 2
\][/tex]
Step 4: Substitute the value of [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex]
Now substitute [tex]\( x = 2 \)[/tex] back into the equation we found for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-2 + 9}{2}
\][/tex]
Simplify the numerator:
[tex]\[
y = \frac{7}{2}
\][/tex]
[tex]\[
y = 3.5
\][/tex]
So the solution to the system of equations is:
[tex]\[
(2, 3.5)
\][/tex]
