You have a system of two linear equations:

1. $3x + 3y = y + 1$
2. $x + 3y = 9 - x$

Let's solve this system step by step.

Step 1: Simplify both equations.

Start with the first equation: $3x + 3y = y + 1$ Subtract $y$ from both sides: $3x + 2y = 1 \quad \text{(Equation 1)}$

Now simplify the second equation: $x + 3y = 9 - x$ Add $x$ to both sides: $2x + 3y = 9 \quad \text{(Equation 2)}$

Step 2: Solve the system using substitution or elimination.

Let's use elimination. We have:

1. $3x + 2y = 1$
2. $2x + 3y = 9$

Multiply the first equation by 2 and the second equation by 3 to align the coefficients of $x$: $2(3x + 2y) = 2(1) \quad \Rightarrow \quad 6x + 4y = 2$ $3(2x + 3y) = 3(9) \quad \Rightarrow \quad 6x + 9y = 27$

Step 3: Subtract the equations.

Subtract the first from the second: $(6x + 9y) - (6x + 4y) = 27 - 2$ Simplifying: $5y = 25$ Solve for $y$: $y = 5$

Step 4: Substitute $y = 5$ back into one of the original equations.

Using Equation 1: $3x + 2(5) = 1$ $3x + 10 = 1$ Subtract 10 from both sides: $3x = -9$ Solve for $x$: $x = -3$

Final Answer:

The solution to the system of equations is: $x = -3, \quad y = 5$

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Would you like further details or have any questions about the solution?

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Here are 5 related questions you could explore:

1. How do you solve systems of linear equations using substitution?
2. What happens if the system has no solution?
3. How do you graph the solution of a system of linear equations?
4. What is the elimination method in solving linear systems?
5. How do you check the solution of a system of equations?

Tip: Always check your solution by substituting the values of $x$ and $y$ back into the original equations to confirm they satisfy both equations.