Solve Linear Equations By Elimination Method: Step-by-Step

Hey guys! Let's dive into solving systems of linear equations using the elimination method. It might sound intimidating, but trust me, it’s a super useful tool for tackling these kinds of problems. We'll break it down step by step, so you'll be a pro in no time! So, let’s jump right into understanding how the elimination method works and how we can apply it to a specific system of equations. This method is incredibly powerful for finding solutions where two or more equations intersect, giving us a clear understanding of the variables at play.

What is 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 goal is to manipulate the equations so that either the x or y coefficients are opposites (e.g., 3 and -3). When you add the equations together, one variable cancels out, leaving you with a single equation in one variable that you can easily solve. This method is particularly effective when the coefficients of one variable are already the same or simple multiples of each other. By strategically eliminating one variable, we simplify the problem, making it much easier to find the values of both x and y that satisfy both equations simultaneously. This technique is a cornerstone of algebra and provides a robust approach to solving systems of equations.

Steps to Solve by Elimination

To effectively use the elimination method, follow these steps:

1. Align the Equations: Make sure the equations are aligned, with like terms in the same columns (x terms over x terms, y terms over y terms, and constants over constants). Proper alignment is crucial for the next steps to work smoothly. This ensures that when we add or subtract the equations, we are combining like terms, which is essential for eliminating a variable.

2. Multiply (if necessary): Look at the coefficients of x and y. If needed, multiply one or both equations by a constant so that the coefficients of one variable are opposites. For example, if you have 2x in one equation and x in another, you might multiply the second equation by -2. This step is often the key to setting up the elimination. By making the coefficients of one variable opposites, we ensure that they will cancel out when the equations are added together, leaving us with a single equation in one variable.

3. Add the Equations: Add the equations together. One variable should cancel out, leaving you with a single equation in one variable. This is the core of the elimination process. When the equations are added, the opposing coefficients eliminate each other, simplifying the system into a more manageable form. The resulting equation can then be solved for the remaining variable.

4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable. This is usually a straightforward algebraic step. Once you've eliminated one variable, you're left with a basic equation that can be easily solved using standard algebraic techniques. This step provides the value of one of the variables in the system.

5. Substitute: Substitute the value you found back into one of the original equations to solve for the other variable. Choose the equation that looks easier to work with. Substituting the known value back into one of the original equations allows you to solve for the remaining unknown variable, completing the solution of the system.

6. Check Your Solution: Plug both values back into both original equations to make sure they are correct. This step is crucial for ensuring accuracy. By verifying the solution in both original equations, you can confirm that the values satisfy the entire system and that no errors were made during the solving process.

Example: Solving the System

Okay, let’s apply these steps to the system you provided:

$$\begin{array}{l}-x-3 y=-17 \\ 3 x+3 y=27\end{array}$$

Step 1: Align the Equations

Good news! The equations are already aligned perfectly:

$$\begin{array}{l}-x-3 y=-17 \\ 3 x+3 y=27\end{array}$$

Step 2: Multiply (if necessary)

Notice that the y coefficients are already opposites (-3 and 3). How convenient is that? We can skip the multiplication step this time. Sometimes, you get lucky, and the problem is set up in a way that minimizes the initial work. In this case, the y coefficients are perfect for elimination, saving us a step.

Step 3: Add the Equations

Add the two equations together:

$$(-x - 3y) + (3x + 3y) = -17 + 27$$

Combine like terms:

$$2x = 10$$

The y terms canceled out, just as we planned! This is the magic of the elimination method – simplifying the system by removing one variable. By adding the equations, we’ve transformed the problem into a single, straightforward equation in terms of x.

Step 4: Solve for the Remaining Variable

Now, solve for x:

$$x = \frac{10}{2} = 5$$

Awesome! We found that x = 5. With one variable solved, we’re halfway there. This value is a crucial piece of the puzzle, and now we can use it to find the value of y. Solving for x is a major milestone in the elimination process, bringing us closer to the complete solution of the system.

Step 5: Substitute

Substitute x = 5 into one of the original equations. Let's use the first equation:

$$-5 - 3y = -17$$

Add 5 to both sides:

$$-3y = -12$$

Divide by -3:

$$y = 4$$

So, y = 4. We now have both x and y values! Substituting the value of x back into the equation allowed us to isolate y and find its corresponding value. This step is essential for completing the solution, as it gives us the value of the second variable in the system.

Step 6: Check Your Solution

Let’s make sure our solution (x = 5, y = 4) is correct by plugging it into both original equations:

First equation:

$$-5 - 3(4) = -5 - 12 = -17$$

Yep, that checks out!

Second equation:

$$3(5) + 3(4) = 15 + 12 = 27$$

Perfect! Our solution is correct. Verification is key to ensure that our solution satisfies both equations in the system. By plugging the values back in, we can confirm that our calculations are accurate and that we have found the correct solution.

Why the Elimination Method is Super Useful

The elimination method is a fantastic tool because:

• It's Efficient: When equations are set up just right (or with a little tweaking), it's a quick way to solve. The elimination method shines when the coefficients align favorably, allowing for a swift elimination of one variable. This efficiency makes it a go-to method for many algebraic problems.
• It Handles Tricky Equations: It works well even when substitution might be a pain. For systems where substitution involves complex fractions or algebraic manipulations, the elimination method often provides a more straightforward path to the solution.
• It's Versatile: You can use it for systems with more than two variables, too! While we’ve focused on two-variable systems, the elimination method can be extended to systems with three or more variables, making it a versatile tool in advanced algebra.

Tips for Success

• Stay Organized: Keep your work neat and tidy. It's easy to make mistakes if your equations and steps are all over the place. A well-organized approach is crucial for accuracy.
• Double-Check: Always check your solution in both original equations. This ensures that your solution is correct and catches any potential errors in your calculations.
• Practice Makes Perfect: The more you practice, the better you'll get at spotting the best way to eliminate variables. Like any skill, mastering the elimination method requires practice. The more problems you solve, the more comfortable and confident you’ll become.

Common Mistakes to Avoid

• Forgetting to Multiply the Entire Equation: If you multiply one side of the equation, make sure you multiply every term on the other side as well. Consistency is key to maintaining the equation's balance.
• Sign Errors: Pay close attention to signs when adding or subtracting equations. A small mistake with a sign can lead to a completely wrong answer. Double-check your work to avoid these common pitfalls.
• Not Checking Your Solution: Always, always check your solution in the original equations. This simple step can save you from submitting an incorrect answer.

Real-World Applications

The elimination method isn't just a math classroom concept; it has practical applications in various fields:

• Engineering: Engineers use systems of equations to design structures, circuits, and more. The elimination method can help solve complex engineering problems by simplifying the underlying equations.
• Economics: Economists use systems of equations to model markets and predict economic trends. The elimination method is a valuable tool for analyzing these models and making informed decisions.
• Computer Science: In computer graphics and simulations, systems of equations are used to calculate transformations and movements. The elimination method can help optimize these calculations for performance.

Conclusion

So there you have it! Using the elimination method to solve systems of linear equations is a powerful and efficient technique. By aligning equations, eliminating variables, and solving step by step, you can conquer these problems with confidence. Remember, practice is key, so keep at it, and you'll become a pro in no time! Solving systems of equations is a fundamental skill in mathematics, and mastering the elimination method will open doors to more advanced concepts and real-world applications.