You have a system of two equations:

1. $x + y = 12$
2. $3x = 2y + 6$

Let's solve this step-by-step.

Step 1: Solve the first equation for one variable.

From the first equation $x + y = 12$, solve for $x$:

$x = 12 - y$

Step 2: Substitute this expression for $x$ into the second equation.

Now, substitute $x = 12 - y$ into the second equation $3x = 2y + 6$:

$3(12 - y) = 2y + 6$

Step 3: Simplify and solve for $y$.

Distribute the 3 on the left side:

$36 - 3y = 2y + 6$

Now, move all terms involving $y$ to one side and constants to the other side:

$36 - 6 = 2y + 3y$ $30 = 5y$

Solve for $y$:

$y = \frac{30}{5} = 6$

Step 4: Substitute $y = 6$ back into the first equation to find $x$.

Now, substitute $y = 6$ into the equation $x = 12 - y$:

$x = 12 - 6 = 6$

Final Answer:

The solution to the system of equations is $x = 6$ and $y = 6$.

Would you like further details on any of the steps, or is there another problem you'd like to tackle?

Related Questions:

1. How would the solution change if the equations were $x + y = 10$ and $3x = 2y + 4$?
2. What happens if the two equations are parallel? How do we know?
3. How can you solve this system using substitution or elimination methods?
4. What is the geometric interpretation of these linear equations on a graph?
5. How do you check the solution for correctness?

Tip: Always check your solution by substituting back into the original equations to verify it satisfies both.