We are given the following system of linear equations:

1. $3x + y = 11$
2. $y = x + 2$

Step 1: Substitute the expression for $y$ from the second equation into the first equation.

From equation (2), we know that $y = x + 2$. We substitute this into equation (1):

$3x + (x + 2) = 11$

Step 2: Simplify and solve for $x$.

$3x + x + 2 = 11$ $4x + 2 = 11$ Now, subtract 2 from both sides: $4x = 9$ Finally, divide both sides by 4: $x = \frac{9}{4}$

Step 3: Substitute the value of $x$ back into the second equation to find $y$.

We know from equation (2) that $y = x + 2$. Substituting $x = \frac{9}{4}$: $y = \frac{9}{4} + 2 = \frac{9}{4} + \frac{8}{4} = \frac{17}{4}$

Final Answer:

The solution to the system of equations is: $x = \frac{9}{4}, \quad y = \frac{17}{4}$

Would you like more details or have any questions about this solution?

Additional questions:

1. How would you solve a system of equations using substitution with fractional coefficients?
2. How can you verify the solution to a system of equations graphically?
3. What is the method for solving systems of equations by elimination?
4. How do you solve a system of linear equations if one equation is already solved for one variable?
5. What is the significance of the solution $\left( \frac{9}{4}, \frac{17}{4} \right)$ in the context of the system?

Tip: When solving by substitution, always ensure that the variable you're substituting into is isolated in one of the equations to make the substitution straightforward!