To solve this system using the substitution method, let’s go through the steps:

Step 1: Solve one equation for one variable

Starting with the second equation: $x + 2y = 1$ Solve for $x$: $x = 1 - 2y$

Step 2: Substitute $x$ in the other equation

Substitute $x = 1 - 2y$ into the first equation: $(1 - 2y) - 3y = -9$ Simplify: $1 - 5y = -9$ Subtract 1 from both sides: $-5y = -10$ Divide by -5: $y = 2$

Step 3: Substitute $y = 2$ back into the equation for $x$

Using $x = 1 - 2y$: $x = 1 - 2(2)$ $x = 1 - 4$ $x = -3$

Solution

The solution to the system is: $(x, y) = (-3, 2)$

Verification

To check graphically, you would plot the lines:

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

They should intersect at the point $(-3, 2)$, confirming our solution.

Would you like a more detailed explanation, or is there something specific you'd like clarified?

Here are some related questions to explore further:

1. How does the substitution method work for systems with no solutions?
2. What are other methods to solve systems of linear equations?
3. Can you use substitution in systems with three variables?
4. What does it mean graphically if two lines in a system are parallel?
5. How can we solve this system by elimination instead of substitution?

Tip: Always check your solution by substituting back into both original equations to ensure accuracy.