We will solve the system of equations using the elimination method.

The system is:

1. $7x + 2y = -18$
2. $7x + 9y = 17$

Step 1: Eliminate one variable

Since both equations have $7x$ terms, we can eliminate $x$ by subtracting one equation from the other.

Subtract equation (1) from equation (2): $(7x + 9y) - (7x + 2y) = 17 - (-18)$ Simplifying both sides: $7x - 7x + 9y - 2y = 17 + 18$ $0 + 7y = 35$ $7y = 35$

Step 2: Solve for $y$

Divide both sides by 7: $y = \frac{35}{7} = 5$

Step 3: Substitute $y = 5$ into one of the original equations

We'll substitute $y = 5$ into equation (1): $7x + 2(5) = -18$ $7x + 10 = -18$ Subtract 10 from both sides: $7x = -18 - 10$ $7x = -28$

Step 4: Solve for $x$

Divide both sides by 7: $x = \frac{-28}{7} = -4$

Final Solution:

$x = -4, \quad y = 5$

Thus, the solution to the system of equations is $x = -4$ and $y = 5$.

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

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5 Related Questions:

1. How would you solve this system using substitution instead of elimination?
2. What if the coefficients of $x$ and $y$ were different? How would the elimination method change?
3. Can you apply elimination to a system with three equations and three unknowns?
4. What happens if we had no solution or infinitely many solutions in the system?
5. How can you check if your solution is correct using substitution?

Tip: Always align the coefficients of the variable you want to eliminate, and make sure the signs are correct when adding or subtracting the equations.