Solving Systems Of Equations: Elimination Method Example

Introduction to Solving Systems of Equations

In mathematics, solving systems of equations is a fundamental skill. A system of equations is a set of two or more equations that involve the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. There are several methods to solve systems of equations, including substitution, graphing, and elimination. In this article, we will focus on the elimination method, also known as the addition method, to solve a system of non-linear equations. Specifically, we will tackle the system:

$$\left\{\begin{array}{l}3 x^2-y^2=11 \ x^2+4 y^2=8\end{array}\right.$$

The elimination method is particularly useful when the coefficients of one of the variables in the equations are opposites or can easily be made opposites by multiplying one or both equations by a constant. This method simplifies the system by eliminating one variable, allowing us to solve for the remaining variable. By understanding and applying the elimination method, you can solve a wide range of systems of equations efficiently and accurately. This approach is not only essential for academic success in algebra and calculus but also has practical applications in various fields such as engineering, economics, and computer science, where solving systems of equations is a common task. The versatility of the elimination method makes it a valuable tool in any mathematician's toolkit. For instance, in physics, it can be used to solve for forces in equilibrium, and in economics, it helps determine market equilibrium by finding the point where supply and demand curves intersect. Therefore, mastering this method provides a solid foundation for more advanced mathematical concepts and real-world problem-solving scenarios.

Understanding the Elimination Method

The elimination method is a technique used to solve systems of equations by adding or subtracting the equations in a way that eliminates one of the variables. This method is particularly effective when the coefficients of one of the variables are either the same or opposites. The goal is to manipulate the equations so that when they are added together, one variable cancels out, leaving a single equation in one variable that can be easily solved. The process generally involves multiplying one or both equations by a constant to make the coefficients of one variable opposites, and then adding the equations to eliminate that variable. Once you've solved for one variable, you can substitute its value back into one of the original equations to solve for the other variable. This step-by-step approach ensures that you find all possible solutions for the system.

To successfully apply the elimination method, it's crucial to ensure that the equations are properly aligned. This means that like terms (e.g., $x^2$ terms, $y^2$ terms, constants) should be vertically aligned. This alignment makes it easier to identify which variable can be eliminated. Furthermore, understanding the underlying principle of maintaining equality is vital. Any operation performed on one side of an equation must also be performed on the other side to keep the equation balanced. This principle applies when multiplying equations by constants or adding equations together. The elimination method is not just a mechanical process; it's a logical approach that leverages the properties of equations to simplify and solve systems. By mastering this method, you gain a powerful tool for tackling a wide range of mathematical problems, including those in higher-level mathematics and applied sciences. Its versatility and efficiency make it a cornerstone of algebraic problem-solving.

Step-by-Step Solution Using Elimination

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

$$\left\{\begin{array}{l}3 x^2-y^2=11 \ x^2+4 y^2=8\end{array}\right.$$

Step 1: Identify a Variable to Eliminate

Observe the coefficients of the variables in both equations. We can eliminate $y^2$ more easily because the coefficients are $-1$ and $4$. To do this, we will multiply the first equation by 4 so that the $y^2$ terms have opposite coefficients.

Step 2: Multiply Equations to Match Coefficients

Multiply the first equation by 4:

$$4(3x^2 - y^2) = 4(11)$$

$$12x^2 - 4y^2 = 44$$

Now our system looks like this:

$$\left\{\begin{array}{l}12x^2 - 4y^2 = 44 \ x^2 + 4y^2 = 8\end{array}\right.$$

Step 3: Add the Equations

Add the two equations together to eliminate the $y^2$ term:

$$(12x^2 - 4y^2) + (x^2 + 4y^2) = 44 + 8$$

$$13x^2 = 52$$

Step 4: Solve for $x$

Divide both sides by 13:

$$x^2 = \frac{52}{13}$$

$$x^2 = 4$$

Take the square root of both sides:

$$x = \pm 2$$

So, we have two possible values for $x$: $x = 2$ and $x = -2$.

Step 5: Substitute $x$ Values to Find $y$

Substitute each value of $x$ into one of the original equations to solve for $y$. Let's use the second equation, $x^2 + 4y^2 = 8$.

For $x = 2$:

$$(2)^2 + 4y^2 = 8$$

$$4 + 4y^2 = 8$$

$$4y^2 = 4$$

$$y^2 = 1$$

$$y = \pm 1$$

So, when $x = 2$, we have two solutions: $(2, 1)$ and $(2, -1)$.

For $x = -2$:

$$(-2)^2 + 4y^2 = 8$$

$$4 + 4y^2 = 8$$

$$4y^2 = 4$$

$$y^2 = 1$$

$$y = \pm 1$$

So, when $x = -2$, we have two solutions: $(-2, 1)$ and $(-2, -1)$.

Step 6: Write the Solutions

The solutions to the system of equations are:

$$(2, 1), (2, -1), (-2, 1), (-2, -1)$$

Common Mistakes to Avoid

When solving systems of equations using the elimination method, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and ensure accurate solutions. One frequent mistake is failing to distribute the multiplication correctly when multiplying an equation by a constant. For example, if you multiply the equation $3x^2 - y^2 = 11$ by 4, you must ensure that every term is multiplied, resulting in $12x^2 - 4y^2 = 44$. Overlooking this distribution can lead to incorrect coefficients and, consequently, wrong solutions.

Another common error occurs when adding or subtracting equations. It's crucial to align like terms properly. This means adding the $x^2$ terms together, the $y^2$ terms together, and the constants together. Mixing up terms can lead to incorrect cancellations and a flawed final equation. Additionally, students sometimes forget to solve for all possible values of a variable, especially when dealing with square roots. For instance, when solving $x^2 = 4$, remember that there are two solutions: $x = 2$ and $x = -2$. Neglecting the negative root can result in missing solutions to the system.

Finally, always double-check your solutions by substituting them back into the original equations. This step is crucial for verifying that the solutions satisfy both equations simultaneously. If a solution doesn't work in both equations, it's not a valid solution. By being mindful of these common mistakes and consistently checking your work, you can improve your accuracy and confidence in solving systems of equations using the elimination method. This meticulous approach not only helps in academic settings but also in real-world applications where precise solutions are essential.

Conclusion

In this article, we successfully used the elimination method to solve a system of non-linear equations. The key steps included multiplying equations to match coefficients, adding the equations to eliminate a variable, solving for the remaining variable, and substituting back to find all solutions. We found that the system $\left\{\begin{array}{l}3 x^2-y^2=11 \ x^2+4 y^2=8\end{array}\right.$ has four solutions: $(2, 1)$, $(2, -1)$, $(-2, 1)$, and $(-2, -1)$.

The elimination method is a powerful tool for solving systems of equations, especially when dealing with equations where variables have coefficients that can be easily manipulated. By mastering this technique, you can tackle a wide range of mathematical problems with confidence. Remember to always check your solutions by substituting them back into the original equations to ensure accuracy. The ability to solve systems of equations is not just an academic exercise; it's a fundamental skill that has applications in various fields, from engineering and physics to economics and computer science. Therefore, continuous practice and understanding of the underlying principles will undoubtedly enhance your problem-solving capabilities.

For further exploration and practice on solving systems of equations, you can visit Khan Academy's Systems of Equations Section.