Certainly! Let's solve the given system of linear equations step-by-step:
[tex]\[
\begin{cases}
x + 2y = 12 \\
y = -3x + 11
\end{cases}
\][/tex]
Step 1: Express one variable in terms of the other.
From the second equation, we already have [tex]\( y \)[/tex] expressed in terms of [tex]\( x \)[/tex]:
[tex]\[
y = -3x + 11
\][/tex]
Step 2: Substitute this expression into the first equation.
Substitute [tex]\( y = -3x + 11 \)[/tex] into the first equation:
[tex]\[
x + 2(-3x + 11) = 12
\][/tex]
Step 3: Simplify and solve for [tex]\( x \)[/tex].
Distribute the 2 through the equation:
[tex]\[
x + 2(-3x) + 2(11) = 12
\][/tex]
[tex]\[
x - 6x + 22 = 12
\][/tex]
Combine like terms:
[tex]\[
-5x + 22 = 12
\][/tex]
Isolate [tex]\( x \)[/tex] by subtracting 22 from both sides:
[tex]\[
-5x = 12 - 22
\][/tex]
[tex]\[
-5x = -10
\][/tex]
Divide both sides by -5:
[tex]\[
x = 2
\][/tex]
Step 4: Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex].
Now that we have [tex]\( x = 2 \)[/tex], substitute it into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = -3(2) + 11
\][/tex]
[tex]\[
y = -6 + 11
\][/tex]
[tex]\[
y = 5
\][/tex]
Step 5: State the solution.
The solution to the system of equations is:
[tex]\[
(x, y) = (2, 5)
\][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = 2 \)[/tex] and [tex]\( y = 5 \)[/tex].
