To solve the system of equations using the elimination method, we want to eliminate one of the variables by adding or subtracting the equations. Here’s how we can do it step-by-step:

Given the system of equations:

1. $-3x - 2y = -20$
2. $4x + 10y = 12$

Step 1: Eliminate one variable

We'll try to eliminate $x$ by making the coefficients of $x$ in both equations equal in magnitude. We can do this by multiplying the first equation by 4 and the second equation by 3, so that both equations will have a coefficient of $-12x$ and $12x$, respectively.

Multiply the first equation by 4: $4(-3x - 2y) = 4(-20)$ This gives: $-12x - 8y = -80$

Multiply the second equation by 3: $3(4x + 10y) = 3(12)$ This gives: $12x + 30y = 36$

Now we have:

1. $-12x - 8y = -80$
2. $12x + 30y = 36$

Step 2: Add the two equations

Now, we'll add these two equations together to eliminate $x$: $(-12x + 12x) + (-8y + 30y) = -80 + 36$ This simplifies to: $0x + 22y = -44$ $22y = -44$

Step 3: Solve for $y$

Now, solve for $y$: $y = \frac{-44}{22} = -2$

Step 4: Substitute $y = -2$ into one of the original equations

Substitute $y = -2$ into one of the original equations to solve for $x$. Let’s use the first equation: $-3x - 2(-2) = -20$ $-3x + 4 = -20$ $-3x = -20 - 4$ $-3x = -24$ $x = \frac{-24}{-3} = 8$

Final Answer:

The solution to the system of equations is: $x = 8, \quad y = -2$

Would you like any further details on this process, or do you have any questions?

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Here are five related questions for further learning:

1. How would you solve this system using substitution instead of elimination?
2. How can you check if this solution is correct?
3. What happens if the two equations represent parallel lines?
4. How do you handle systems with no solution or infinitely many solutions?
5. What is the geometric interpretation of the elimination method in solving systems of equations?

Tip: Always check your final solution by substituting the values of $x$ and $y$ back into the original equations!