The system of equations given in the image is:

$-2x - 3y = -3$ $10x + 8y = -6$

To solve this system, we can use either the elimination or substitution method. Here, I will proceed with the elimination method by eliminating $x$.

Step 1: Make the coefficients of $x$ equal

To eliminate $x$, let's make the coefficients of $x$ in both equations the same. We can do this by multiplying the first equation by $5$ (since $5 \times -2x = -10x$).

Multiplying the first equation by $5$:

$5(-2x - 3y) = 5(-3)$ $-10x - 15y = -15$

Now we have: $-10x - 15y = -15$ $10x + 8y = -6$

Step 2: Add the equations

Now we add both equations together to eliminate $x$:

$(-10x - 15y) + (10x + 8y) = -15 + (-6)$ $0x - 7y = -21$ $-7y = -21$

Step 3: Solve for $y$

Now, we solve for $y$ by dividing both sides by $-7$:

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

Step 4: Substitute $y = 3$ back into one of the original equations

We can substitute $y = 3$ into the first equation to find $x$:

$-2x - 3(3) = -3$ $-2x - 9 = -3$ $-2x = -3 + 9$ $-2x = 6$ $x = \frac{6}{-2}$ $x = -3$

Solution

The solution to the system is: $x = -3, \quad y = 3$

Would you like further explanation or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the solution change if we used the substitution method instead?
2. What does it mean graphically when we find a solution to a system of equations?
3. Can a system of equations have no solution? If so, how would that look in this example?
4. How would you verify the solution for this system of equations?
5. What steps would you take if both equations had fractions?

Tip: When solving systems of equations, choosing the method (elimination or substitution) that involves the least amount of calculation often makes the process faster and easier.