The given system of equations is:

$9x + 2y = 19$ $6x + 7y = -10$

We will solve this system using substitution or elimination. Let's proceed by using the elimination method.

Step 1: Eliminate one variable

To eliminate one of the variables, we'll make the coefficients of either $x$ or $y$ the same in both equations. Let's focus on eliminating $x$.

Multiply the first equation by 6 and the second equation by 9 to align the coefficients of $x$:

$6(9x + 2y) = 6(19) \quad \Rightarrow \quad 54x + 12y = 114$ $9(6x + 7y) = 9(-10) \quad \Rightarrow \quad 54x + 63y = -90$

Step 2: Subtract the equations

Now subtract the second equation from the first:

$(54x + 12y) - (54x + 63y) = 114 - (-90)$ $54x + 12y - 54x - 63y = 114 + 90$ $-51y = 204$

Step 3: Solve for $y$

$y = \frac{204}{-51} = -4$

Step 4: Substitute $y = -4$ into one of the original equations

We substitute $y = -4$ into the first equation:

$9x + 2(-4) = 19$ $9x - 8 = 19$ $9x = 19 + 8 = 27$ $x = \frac{27}{9} = 3$

Final Answer:

$x = 3, \quad y = -4$

Would you like further details or explanations on any step?

Here are some additional questions that may help expand your understanding:

1. How would you solve this system using the substitution method?
2. How can the determinant method (Cramer's Rule) be used to solve this system of equations?
3. What happens if both equations are multiplied to eliminate $y$ instead of $x$?
4. Can graphing both equations provide a visual confirmation of the solution?
5. How would solving the system change if one of the constants was different?

Tip: When using the elimination method, try to choose the variable that simplifies the arithmetic the most.