Let's solve the system of equations using the substitution method:

Given:

1. $x - y = 11$
2. $2x + y = 19$

Step 1: Solve one equation for one variable in terms of the other

From the first equation $x - y = 11$, solve for $x$ in terms of $y$: $x = y + 11$

Step 2: Substitute the expression for $x$ in the second equation

Now, substitute $x = y + 11$ into the second equation $2x + y = 19$: $2(y + 11) + y = 19$

Step 3: Simplify and solve for $y$

Distribute the 2: $2y + 22 + y = 19$ Combine like terms: $3y + 22 = 19$ Subtract 22 from both sides: $3y = -3$ Divide by 3: $y = -1$

Step 4: Substitute $y = -1$ back into the equation for $x$

Now, use $x = y + 11$ and substitute $y = -1$: $x = -1 + 11 = 10$

Final Solution

The solution to the system of equations is: $x = 10, \quad y = -1$

Would you like further clarifications, or is there another step you'd like me to explore?

Here are 5 related questions to enhance your understanding:

1. What would happen if we used the elimination method instead of substitution?
2. How can we verify if our solution is correct?
3. Can every system of linear equations be solved using substitution? If not, why?
4. What are other methods to solve systems of equations?
5. How do substitution and elimination methods compare in efficiency for larger systems?

Tip: Always substitute your solutions back into the original equations to confirm they satisfy both equations.