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The elimination method is a popular technique to solve systems of linear equations. The goal is to eliminate one of the variables, making it easier to solve for the other. Here's how it works:

To use elimination, we think about combining the equations in such a way that one of the variables cancels out.
This often involves adding or subtracting the equations from each other. For example, in the given system

• 3x - y = 9
• x + y = -1

we can add the two equations. Notice how the "+y" and "-y" terms will cancel each other. The result is a new equation with only one variable: 4x = 8.

With one variable gone, finding the value of x or y becomes straightforward. This method is efficient and reduces the complexity of the system. Once we find one variable, we plug it back into one of the original equations to solve for the other variable. This makes the elimination method very powerful for solving linear equations quickly.

Solving linear equations is all about finding values for variables that make the equation true.
Linear equations typically take the form of ax + by = c. Each letter represents a constant or coefficient that helps define the relationship between x and y.

In the elimination method example, once we isolated the equation 4x = 8, we solved for x by dividing both sides by 4, which gives us x = 2.

• Begin by isolating one of the variables on one side of the equation, if needed.
• In this case, we already have 4x on one side, so dividing by 4 simplifies it further.

Once x is determined, solving for y involves substituting x back into one of the original equations. In this example, substituting x = 2 into x + y = -1 helps us find y = -3. Solving linear equations involves manipulating the equations carefully to find accurate values for all variables involved.

Once we have proposed solutions for x and y, the next step is to verify these values to ensure they satisfy the original equations. This step, known as checking solutions, is crucial to confirm our solutions are correct.

We substitute the values of x and y back into both original equations:

• For 3x - y = 9, substituting x = 2 and y = -3 gives us 3(2) - (-3) = 9, which simplifies to 6 + 3 = 9. Hence, the equation holds true.
• For x + y = -1, substituting x = 2 and y = -3 gives 2 - 3 = -1, confirming this equation is also correct.

Checking solutions ensures that our arithmetic is correct and the solutions make sense in the context of the problem. It validates that both equations in the system are satisfied with the obtained values of x and y. This final step gives confidence in the accuracy of our solutions and reinforces understanding of the solving process.