Let's solve this system of equations using the elimination method.

Equations:

1. $4x + 3y = -2$
2. $8x - 2y = 12$

Step 1: Multiply the equations to align coefficients for elimination.

We aim to eliminate one variable by multiplying the equations so that one of the variables has the same or opposite coefficients. Let's eliminate $x$. Notice that the coefficients of $x$ in the two equations are 4 and 8, so we can multiply Equation 1 by 2 to match the coefficient of $x$ in Equation 2:

$2(4x + 3y) = 2(-2)$ $8x + 6y = -4 \quad \text{(new Equation 1)}$

Now, we have the system:

1. $8x + 6y = -4$
2. $8x - 2y = 12$

Step 2: Subtract the equations to eliminate $x$.

Now subtract Equation 2 from the modified Equation 1 to eliminate $x$:

$(8x + 6y) - (8x - 2y) = -4 - 12$

Simplifying:

$8x - 8x + 6y + 2y = -16$ $0x + 8y = -16$ $8y = -16$

Step 3: Solve for $y$.

Divide both sides by 8:

$y = \frac{-16}{8} = -2$

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

Substitute $y = -2$ into the first original equation $4x + 3y = -2$:

$4x + 3(-2) = -2$ $4x - 6 = -2$ $4x = 4$ $x = 1$

Final Answer:

• $x = 1$
• $y = -2$

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Would you like further details on any step, or have any additional questions? Here are some related questions to explore further:

1. How would the solution differ if we used substitution instead of elimination?
2. Can elimination be used for systems with more than two equations?
3. What happens if the two lines are parallel? How would elimination work in that case?
4. How can we check if our solution is correct using substitution or graphing?
5. What do we do if one of the variables cancels out completely in elimination?

Tip: When solving by elimination, always check if multiplying one or both equations by a factor will align the variables for easy cancellation.