The problem involves solving the system of equations using the elimination method. The given system of equations is:

$2x + 7y = -10$ $6x + 5y = 34$

Steps to Solve Using Elimination:

1. Align the equations for elimination by eliminating one variable (e.g., $x$): To do this, multiply both equations so that the coefficients of $x$ are equal. Multiply the first equation by 3 and the second equation by 1: $3(2x + 7y) = 3(-10) \implies 6x + 21y = -30$ $6x + 5y = 34$

   The system now becomes: $6x + 21y = -30$ $6x + 5y = 34$

2. Subtract the equations to eliminate $x$: $(6x + 21y) - (6x + 5y) = -30 - 34$ Simplify: $16y = -64$

3. Solve for $y$: $y = \frac{-64}{16} = -4$

4. Substitute $y = -4$ into one of the original equations (e.g., $2x + 7y = -10$): $2x + 7(-4) = -10$ Simplify: $2x - 28 = -10$ $2x = 18$ $x = 9$

Final Solution:

$x = 9, \, y = -4$

Would you like a more detailed explanation or have additional questions?

5 Related Questions:

1. What is substitution, and how does it compare to elimination for solving systems of equations?
2. How would the solution change if the second equation was $6x + 5y = 40$?
3. Can a system of equations have no solution? If so, how can we identify it?
4. What are the steps for graphing these equations to verify the solution?
5. How would elimination work for systems with three variables and three equations?

Tip:

When using elimination, always double-check the arithmetic when combining or subtracting equations to avoid errors.