Sure! Let's solve the given system of equations step-by-step:
[tex]\[
\begin{cases}
\frac{2(x+1)}{3} + 3 = y \\
3(x + 5 - y) + 3x = 12
\end{cases}
\][/tex]
### Step 1: Simplify the First Equation
First, we simplify the first equation:
[tex]\[
\frac{2(x+1)}{3} + 3 = y
\][/tex]
Multiply both sides by 3 to clear the fraction:
[tex]\[
2(x+1) + 9 = 3y
\][/tex]
Distribute and simplify:
[tex]\[
2x + 2 + 9 = 3y \\
2x + 11 = 3y
\][/tex]
Then solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{2x + 11}{3}
\][/tex]
### Step 2: Substitute [tex]\(y\)[/tex] into the Second Equation
Now we substitute [tex]\(y = \frac{2x + 11}{3}\)[/tex] into the second equation:
[tex]\[
3(x + 5 - y) + 3x = 12
\][/tex]
Substitute [tex]\(y\)[/tex]:
[tex]\[
3(x + 5 - \frac{2x + 11}{3}) + 3x = 12
\][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Clear the fraction by multiplying every term by 3:
[tex]\[
3 \cdot 3(x + 5) - 3 \cdot (2x + 11) + 9x = 36
\][/tex]
Simplify and distribute:
[tex]\[
3(3x + 15) - 2(2x + 11) = 36 \\
9x + 45 - 2(2x + 11) = 36
\][/tex]
Now, distribute [tex]\( -2 \)[/tex] and simplify:
[tex]\[
9x + 45 - 4x - 22 = 36 \\
5x + 23 = 36
\][/tex]
Subtract 23 from both sides:
[tex]\[
5x = 13
\][/tex]
Divide by 5:
[tex]\[
x = \frac{13}{5}
\][/tex]
### Step 4: Solve for [tex]\(y\)[/tex]
Now we find [tex]\(y\)[/tex] using [tex]\(y = \frac{2x + 11}{3}\)[/tex] with [tex]\(x = \frac{13}{5}\)[/tex]:
[tex]\[
y = \frac{2 \left(\frac{13}{5}\right) + 11}{3}
\][/tex]
Multiply the numerator inside the fraction:
[tex]\[
y = \frac{\frac{26}{5} + 11}{3}
\][/tex]
Convert 11 into a fraction with a common denominator:
[tex]\[
y = \frac{\frac{26}{5} + \frac{55}{5}}{3} \\
y = \frac{\frac{81}{5}}{3}
\][/tex]
Simplify:
[tex]\[
y = \frac{81}{15}
\][/tex]
Further simplify:
[tex]\[
y = \frac{27}{5}
\][/tex]
So the solutions are:
[tex]\[
x = 2, \quad y = 5
\][/tex]
This implies that [tex]\( (x, y) = (2, 5) \)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[
\boxed{(2, 5)}
\][/tex]
