Solving Systems Of Equations By Elimination A Step-by-Step Guide

In mathematics, solving a system of equations is a fundamental skill with applications across various fields, from engineering and physics to economics and computer science. One powerful method for tackling these systems is elimination, a technique that systematically combines equations to eliminate variables and simplify the problem. This article provides a comprehensive guide on how to solve systems of equations using elimination, walking you through the process step-by-step with clear explanations and examples. We'll focus on the given system of equations:

$$\begin{array}{l} 3x + 2y = -3 \\ 9x + 4y = 3 \end{array}$$

and determine the correct solution from the provided options: A) $x=3, y=6$, B) $x=-6, y=3$, C) $x=-6, y=-3$, and D) $x=3, y=-6$. By the end of this guide, you'll not only be able to solve this specific problem but also confidently apply the elimination method to a wide range of systems of equations.

Understanding the Elimination Method

The elimination method, also known as the addition method, is a technique used to solve systems of linear equations by eliminating one of the variables. The core idea is to manipulate the equations so that the coefficients of one variable are opposites (i.e., additive inverses). When the equations are added together, this variable cancels out, leaving a single equation with one variable, which can then be easily solved. Once one variable is found, its value is substituted back into one of the original equations to solve for the other variable. This method is particularly effective when dealing with systems of equations where the coefficients of one variable are multiples of each other or can be easily made so. The elimination method is a cornerstone of algebraic problem-solving, providing a structured approach to finding solutions in various mathematical and real-world contexts. Its efficiency and clarity make it an indispensable tool for students and professionals alike.

Steps Involved in Elimination

The process of solving systems of equations by elimination involves a series of well-defined steps. First, it's crucial to align the equations, ensuring that like terms (terms with the same variable) are vertically aligned. This setup makes it easier to identify the variables you want to eliminate. Next, the key step is to multiply one or both equations by a constant. The goal here is to make the coefficients of one of the variables opposites. For instance, if one equation has a $2x$ term and the other has a $3x$ term, you might multiply the first equation by 3 and the second by -2 to get $6x$ and $-6x$, respectively. Once you have opposite coefficients, add the equations together. This step should eliminate one variable, leaving you with a single equation in one variable. Solve this resulting equation to find the value of one variable. Finally, substitute the value you found back into one of the original equations and solve for the other variable. This process provides a systematic way to find the values of all variables in the system. Let's get into a practical example for better understanding.

Applying Elimination to the Given System

Let's apply the elimination method to the given system of equations:

$$\begin{array}{l} 3x + 2y = -3 \\ 9x + 4y = 3 \end{array}$$

Step 1: Align the Equations

The equations are already aligned, with the $x$ terms, $y$ terms, and constants in their respective columns. This alignment is crucial for the next steps in the elimination process.

Step 2: Multiply to Obtain Opposite Coefficients

To eliminate a variable, we need to make the coefficients of either $x$ or $y$ opposites. Let's choose to eliminate $x$. Notice that the coefficient of $x$ in the second equation (9) is a multiple of the coefficient of $x$ in the first equation (3). We can multiply the first equation by -3 to make the coefficients of $x$ opposites:

$$-3(3x + 2y) = -3(-3)$$

This simplifies to:

$$-9x - 6y = 9$$

Now our system of equations looks like this:

$$\begin{array}{l} -9x - 6y = 9 \\ 9x + 4y = 3 \end{array}$$

Step 3: Add the Equations

Adding the two equations together, we get:

$$(-9x - 6y) + (9x + 4y) = 9 + 3$$

Simplifying, the $x$ terms cancel out:

$$-2y = 12$$

Step 4: Solve for $y$

Divide both sides by -2 to solve for $y$:

$$y = \frac{12}{-2} = -6$$

So, we have found that $y = -6$.

Step 5: Substitute to Solve for $x$

Substitute the value of $y$ back into one of the original equations. Let's use the first equation:

$$3x + 2y = -3$$

Substitute $y = -6$:

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

Simplify:

$$3x - 12 = -3$$

Add 12 to both sides:

$$3x = 9$$

Divide by 3:

$$x = 3$$

Thus, we have found the solution $x = 3$ and $y = -6$.

Verifying the Solution

To ensure our solution is correct, it's crucial to verify the values of $x$ and $y$ in both original equations. This step confirms that the solution satisfies both equations simultaneously, which is the hallmark of a correct solution to a system of equations. Substituting the values back into the original equations acts as a check against potential errors made during the elimination process or subsequent algebraic manipulations. Let's take the solution we found, $x = 3$ and $y = -6$, and substitute these values into each of the original equations to confirm their validity. This process not only boosts confidence in the solution but also reinforces the understanding of what it means to solve a system of equations.

Verification Process

Let's substitute $x = 3$ and $y = -6$ into the first equation:

$$3x + 2y = -3$$

$$3(3) + 2(-6) = -3$$

$$9 - 12 = -3$$

$$-3 = -3$$

The first equation holds true. Now, let's substitute the values into the second equation:

$$9x + 4y = 3$$

$$9(3) + 4(-6) = 3$$

$$27 - 24 = 3$$

$$3 = 3$$

The second equation also holds true. Since the values satisfy both equations, our solution is correct.

The Correct Solution

Based on our calculations and verification, the correct solution to the system of equations is $x = 3$ and $y = -6$. This corresponds to option D in the given choices.

Conclusion

In this article, we've demonstrated how to solve a system of linear equations using the elimination method. We walked through each step, from aligning the equations and manipulating coefficients to solving for the variables and verifying the solution. The key takeaway is that elimination is a powerful technique for solving systems of equations, particularly when coefficients can be easily manipulated to create opposites. By mastering this method, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications that involve systems of equations. Remember to practice consistently and apply these steps methodically to ensure accuracy and efficiency in your problem-solving.

By carefully following the steps outlined in this guide, you can confidently solve systems of equations using the elimination method. This skill is essential for various mathematical and real-world applications. The elimination method provides a systematic and efficient way to find solutions, making it a valuable tool in your mathematical toolkit. The solution to the given system of equations is $x = 3$ and $y = -6$, which corresponds to option D.