It appears you're asking for guidance on using the elimination method to solve a system of equations in two variables, where the coefficients of the variables may be oppositely equal or have some other relationship. Here's how to approach it based on your description:
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If the variables have opposite coefficients:
- Identify two equations in the system. If the coefficients of one variable are opposites (e.g., \( +3x \) in one equation and \( -3x \) in another equation), you can simply add the two equations together. This will eliminate that variable.
- Example: For the equations \(3x + 4y = 10\) and \(-3x + 2y = 6\): \[ (3x + 4y) + (-3x + 2y) = 10 + 6 \] Which simplifies to \(6y = 16\).
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If the variables have equal coefficients:
- When the coefficients of one variable are equal (e.g., \( +2x \) in one equation and \( +2x \) in another), you can subtract one equation from the other in order to eliminate that variable.
- Example: For the equations \(2x + 3y = 8\) and \(2x - y = 4\): \[ (2x + 3y) - (2x - y) = 8 - 4 \] This simplifies to \(4y = 4\).
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Solve for the Remaining Variable: Once you eliminate one variable, solve for the other variable.
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Back Substitute: After finding one variable, substitute it back into one of the original equations to find the other variable.
In both cases, your goal is to simplify and solve the system by eliminating variables strategically based on their coefficients.
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