Certainly! Let's solve this system of linear equations step by step:
We have the following system of equations:
[tex]\[
\begin{cases}
x + 2y = 12 \\
y = -3x + 11
\end{cases}
\][/tex]
Step 1: Substitute the second equation into the first equation
We know from the second equation that [tex]\( y = -3x + 11 \)[/tex]. We can substitute this expression for [tex]\( y \)[/tex] in the first equation:
[tex]\[
x + 2(-3x + 11) = 12
\][/tex]
Step 2: Simplify the equation
Distribute the 2 into the expression [tex]\(-3x + 11\)[/tex]:
[tex]\[
x - 6x + 22 = 12
\][/tex]
Combine like terms:
[tex]\[
-5x + 22 = 12
\][/tex]
Step 3: Solve for [tex]\( x \)[/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: Solve for [tex]\( y \)[/tex]
Use the value of [tex]\( x \)[/tex] that we found to determine [tex]\( y \)[/tex]. Substitute [tex]\( x = 2 \)[/tex] back into the second original equation [tex]\( y = -3x + 11 \)[/tex]:
[tex]\[
y = -3(2) + 11
\][/tex]
Simplify:
[tex]\[
y = -6 + 11
\][/tex]
[tex]\[
y = 5
\][/tex]
Solution:
The solution to the system of equations is:
[tex]\[
(x, y) = (2, 5)
\][/tex]
