To solve the system of linear equations using the substitution method, we will follow these steps:

Given system of equations:

1. $x - 3y = -9$
2. $2x + 7y = 8$

Step 1: Solve one equation for one variable

Let's solve the first equation for $x$ in terms of $y$.

$x - 3y = -9$

Add $3y$ to both sides:

$x = 3y - 9$

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

Now substitute $x = 3y - 9$ into the second equation:

$2x + 7y = 8$

Substitute $x$:

$2(3y - 9) + 7y = 8$

Step 3: Solve for $y$

Expand the equation:

$6y - 18 + 7y = 8$

Combine like terms:

$13y - 18 = 8$

Add 18 to both sides:

$13y = 26$

Now divide by 13:

$y = 2$

Step 4: Substitute $y = 2$ back into the expression for $x$

Substitute $y = 2$ into $x = 3y - 9$:

$x = 3(2) - 9 = 6 - 9 = -3$

Step 5: Solution

The solution to the system is:

$x = -3, \quad y = 2$

Final Answer:

The solution to the system of equations is $x = -3$ and $y = 2$.

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Do you want further details or have any questions?

Here are 5 related questions to explore:

1. How can we verify if $x = -3$ and $y = 2$ satisfy the original equations?
2. How would you solve this system using the elimination method?
3. What are some real-world applications of solving systems of linear equations?
4. Can a system of linear equations have no solution? How do you know?
5. What is the difference between consistent, inconsistent, and dependent systems of equations?

Tip: When using substitution, always simplify expressions carefully before substituting values back into other equations.