Solve the system of equations using elimination: -8x - 8y = 0 and -5x - 7y = -12.

Understand the Problem

The question is asking us to solve a system of equations using the elimination method. It provides two equations: -8x - 8y = 0 and -5x - 7y = -12. We will need to manipulate these equations to eliminate one variable and solve for the other.

Answer

The solution to the system is \( (x, y) = (-6, 6) \).

Steps to Solve

Rearranging Equations

We start with the two equations:

[ -8x - 8y = 0 \tag{1} ] [ -5x - 7y = -12 \tag{2} ]

Let's simplify equation (1) by dividing everything by -8:

[ x + y = 0 \tag{3} ]

Express one variable in terms of the other

From equation (3):

[ y = -x \tag{4} ]

Now we can substitute (y) in equation (2).

Substituting into the second equation

Substituting equation (4) into equation (2):

[ -5x - 7(-x) = -12 ]

This simplifies to:

[ -5x + 7x = -12 ]

Combining like terms

Combining the terms gives:

[ 2x = -12 ]

Solving for (x)

Dividing both sides by 2:

[ x = -6 ]

Finding (y)

Now substitute (x = -6) back into equation (4):

[ y = -(-6) = 6 ]

Thus, we have:

[ x = -6 \quad \text{and} \quad y = 6 ]

The solution to the system of equations is ( (x, y) = (-6, 6) ).

More Information

In this system of equations, we used elimination to express one variable in terms of the other, making it easier to solve for each variable step by step. Understanding the relationship between the equations is key to finding the solution.

Tips

Forgetting to simplify equations before substituting can lead to mistakes.
Mixing up signs during addition or subtraction when combining like terms.
Not back-substituting the value of one variable into the correct equation.

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