Let's solve the given system of equations step-by-step:
[tex]\[
\begin{array}{l}
1) \, 3x + y = 2x + 3y \\
2) \, 2x + 3y = x + y + 44 \\
\end{array}
\][/tex]
Step 1: Solve the first equation
[tex]\[ 3x + y = 2x + 3y \][/tex]
Subtract [tex]\(2x\)[/tex] and [tex]\(y\)[/tex] from both sides:
[tex]\[ 3x - 2x + y - y = 2x - 2x + 3y - y \][/tex]
This simplifies to:
[tex]\[ x = 2y \][/tex]
Step 2: Substitute [tex]\( x = 2y \)[/tex] into the second equation
[tex]\[ 2x + 3y = x + y + 44 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( 2y \)[/tex]:
[tex]\[ 2(2y) + 3y = 2y + y + 44 \][/tex]
Simplify:
[tex]\[ 4y + 3y = 2y + y + 44 \][/tex]
Combine like terms:
[tex]\[ 7y = 3y + 44 \][/tex]
Subtract [tex]\( 3y \)[/tex] from both sides:
[tex]\[ 7y - 3y = 44 \][/tex]
This simplifies to:
[tex]\[ 4y = 44 \][/tex]
Divide both sides by 4:
[tex]\[ y = 11 \][/tex]
Step 3: Substitute [tex]\( y = 11 \)[/tex] back into [tex]\( x = 2y \)[/tex]
[tex]\[ x = 2(11) \][/tex]
This simplifies to:
[tex]\[ x = 22 \][/tex]
Solution:
The values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are:
[tex]\[ x = 22 \][/tex]
[tex]\[ y = 11 \][/tex]
Thus, the solution to the system of equations is [tex]\( (22, 11) \)[/tex].
