To solve the given system of equations using the elimination method, follow these steps:

Given Equations:

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

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Step 1: Align and prepare for elimination

Multiply the first equation by $3$ so that the coefficients of $x$ in both equations can be eliminated when subtracted:

$3(x + 2y) = 3(-12)$ $3x + 6y = -36 \quad \text{(new equation 1)}$

Now the system is:

1. $3x + 6y = -36$
2. $3x - 10y = 12$

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Step 2: Eliminate $x$

Subtract the second equation from the first:

$(3x + 6y) - (3x - 10y) = -36 - 12$ $3x - 3x + 6y + 10y = -48$ $16y = -48$

Solve for $y$: $y = \frac{-48}{16} = -3$

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

Use the first equation: $x + 2y = -12$ $x + 2(-3) = -12$ $x - 6 = -12$ $x = -12 + 6$ $x = -6$

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Final Solution:

$x = -6, \, y = -3$

The solution to the system is: $\boxed{(-6, -3)}$

Would you like a graphical representation or further details?

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Questions to Expand Understanding:

1. What are the advantages of using elimination over substitution for solving systems of equations?
2. How would the solution process change if the coefficients required scaling both equations?
3. Can this system of equations be solved graphically, and how would the solution look on a graph?
4. What real-life scenarios could involve solving such a system of equations?
5. How can we verify the solution for accuracy after solving?

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Tip:

Always double-check your calculations by substituting the solution back into the original equations to ensure both are satisfied.