We are given the system of linear equations:

$-9x + 7y = 7 \quad \text{(1)}$ $6x - 7y = 14 \quad \text{(2)}$

Let's solve this system using the elimination method.

Step 1: Add the two equations

Notice that the $7y$ terms have opposite signs, so adding the equations will eliminate $y$.

$(-9x + 7y) + (6x - 7y) = 7 + 14$

Simplifying this:

$-9x + 6x = 21$ $-3x = 21$

Step 2: Solve for $x$

Divide both sides by $-3$:

$x = \frac{21}{-3} = -7$

Step 3: Substitute $x = -7$ into one of the original equations

Substitute into equation (2) to solve for $y$:

$6(-7) - 7y = 14$ $-42 - 7y = 14$

Add $42$ to both sides:

$-7y = 14 + 42$ $-7y = 56$

Step 4: Solve for $y$

Divide both sides by $-7$:

$y = \frac{56}{-7} = -8$

Final Solution:

The solution to the system of equations is: $x = -7, \quad y = -8$

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would you solve this system using substitution?
2. Can you verify the solution by substituting into both original equations?
3. What are the general steps in the elimination method for solving systems of equations?
4. How does this method differ from solving systems graphically?
5. What is the geometric interpretation of solving a system of two linear equations?

Tip: When using the elimination method, look for variables with coefficients that can easily cancel out!