Master Linear Equations: Elimination Method Explained

Welcome, math enthusiasts, to a deep dive into one of the most powerful techniques for solving systems of linear equations: the elimination method. This method is your secret weapon when faced with equations that don't immediately lend themselves to substitution. We'll tackle a specific problem, breaking down each step to ensure you feel confident and capable. Our goal is to transform potentially tricky equations into a clear path towards a solution, making algebra less daunting and more manageable. Get ready to unlock the secrets of algebraic manipulation and discover how to efficiently find the values of variables that satisfy multiple equations simultaneously.

Understanding the Elimination Method

The elimination method is a strategy used to solve systems of linear equations by manipulating the equations so that one of the variables cancels out, or is "eliminated," when the equations are added or subtracted. This leaves you with a single equation containing only one variable, which is much easier to solve. Think of it as a strategic dance where you carefully arrange the steps so that two dancers exit the stage at the same time, leaving just one to take a bow. The beauty of elimination lies in its elegance and efficiency, particularly when the coefficients of one variable are opposites or can easily be made opposites. It's a systematic approach that relies on the fundamental properties of equality – whatever you do to one side of an equation, you must do to the other to maintain the balance. This principle is key to ensuring that the solutions you find are valid for the original system. We'll explore how to add, subtract, and multiply equations to achieve the desired cancellation, paving the way for a straightforward solution. This method is especially useful when dealing with equations where isolating a variable for substitution might lead to fractions or more complex expressions. By aiming to eliminate a variable directly, we often simplify the problem considerably. The core idea is to create a situation where adding or subtracting the two equations results in the disappearance of either the 'x' or the 'y' term, leaving you with a simple linear equation in the remaining variable. Once you solve for that variable, you can substitute its value back into one of the original equations to find the value of the other variable, thereby solving the entire system.

Setting Up the Equations for Elimination

Before we can eliminate a variable, our equations need to be in a standard form, typically $Ax + By = C$. This means rearranging them so that the $x$ terms are aligned, the $y$ terms are aligned, and the constant terms are on the other side of the equals sign. Let's look at our given system:

Equation 1: $3x - 30 = y$ Equation 2: $7y - 6 = 3x$

To get them into the standard form, we'll perform some algebraic "tidying up." For Equation 1, we want the $y$ term on the same side as the $x$ term. Subtracting $y$ from both sides gives us: $3x - y = 30$. Now, for Equation 2, we need to move the $3x$ term to the left side. Subtracting $3x$ from both sides yields: $-3x + 7y = 6$. Our system is now ready for elimination:

Standardized Equation 1: $3x - y = 30$ Standardized Equation 2: $-3x + 7y = 6$

Notice how the $x$ terms have coefficients that are opposites (3 and -3). This is the ideal scenario for elimination by addition. If the coefficients weren't opposites, we would proceed to the next step of multiplying one or both equations to make them opposites, but here, luck is on our side! This initial step of rearranging equations is crucial. It ensures that when we perform operations like addition or subtraction, we are combining like terms correctly, and the fundamental relationships within the system of equations are preserved. It’s like preparing the ingredients before you start cooking; proper preparation leads to a much smoother and more successful outcome. Always double-check your rearrangements to ensure accuracy, as a single mistake here can lead to an incorrect final answer.

Executing the Elimination

Now comes the exciting part: the elimination itself! Since the coefficients of our $x$ terms are already opposites ($+3x$ and $-3x$), we can simply add the two standardized equations together. This is where the magic happens:

   (3x -  y = 30)
+ (-3x + 7y =  6)
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    0x + 6y = 36

As you can see, the $x$ terms ($3x$ and $-3x$) cancel each other out, or