Solving Linear Equations By Elimination A Step-by-Step Guide

by Scholario Team 61 views

Hey guys! Today, we're going to dive into the world of linear equations and explore a powerful method for solving them: elimination. If you've ever felt a bit lost when faced with a system of equations, don't worry – we'll break it down step-by-step so you can tackle these problems with confidence. We'll use a specific example to illustrate the process, but the principles we learn here can be applied to any system of linear equations. So, grab your pencils and let's get started!

Understanding the Elimination Method

The elimination method, also known as the addition method, is a technique used to solve systems of linear equations. The main idea behind it is to manipulate the equations in such a way that when you add them together, one of the variables cancels out, leaving you with a single equation in one variable. This makes it much easier to solve for that variable. Once you've found the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable. It’s like a clever mathematical dance where we carefully adjust the steps to make one dancer disappear, leaving the other in the spotlight!

Why Use Elimination?

You might be wondering, "Why use elimination when there are other methods like substitution?" Well, the elimination method is particularly useful when the coefficients of one of the variables are opposites or multiples of each other. This makes it easier to eliminate a variable with fewer steps. Plus, it's a great way to develop your algebraic skills and deepen your understanding of how equations work. Think of it as adding another tool to your mathematical toolbox – the more tools you have, the better equipped you are to handle any problem that comes your way.

The Steps Involved

The elimination method generally involves the following steps:

  1. Align the Equations: Make sure the equations are written in standard form (Ax + By = C).
  2. Multiply (if necessary): Multiply one or both equations by a constant so that the coefficients of one variable are opposites (or the same).
  3. Add the Equations: Add the equations together. This should eliminate one variable.
  4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
  5. Substitute: Substitute the value you found back into one of the original equations to solve for the other variable.
  6. Check: Check your solution by substituting both values into both original equations.

Sounds like a plan? Let’s put these steps into action with our example!

Our Example System of Equations

Let's consider the following system of linear equations:

8x - 2y = -14
-x - 10y = -29

This is the system we'll be working with today. Our goal is to find the values of x and y that satisfy both equations simultaneously. In other words, we're looking for the point where these two lines intersect if we were to graph them. Think of it as a detective case where we have two clues (the equations) and we need to find the hidden values (the variables) that fit both clues perfectly.

Step-by-Step Solution

Okay, let’s walk through the steps of solving this system using the elimination method.

Step 1: Align the Equations

First, we need to make sure our equations are aligned in the standard form (Ax + By = C). Luckily, our equations are already in this form:

8x - 2y = -14
-x - 10y = -29

This step is like making sure our puzzle pieces are facing the right way before we try to fit them together. It might seem simple, but it’s crucial for setting up the next steps.

Step 2: Multiply (if necessary)

Next, we want to multiply one or both equations by a constant so that the coefficients of either x or y are opposites. Looking at our equations, we can see that the coefficient of x in the first equation is 8, and in the second equation, it's -1. If we multiply the second equation by 8, the coefficients of x will be 8 and -8, which are opposites. This is exactly what we want!

So, let’s multiply the second equation (-x - 10y = -29) by 8:

8 * (-x - 10y) = 8 * (-29)
-8x - 80y = -232

Now our system of equations looks like this:

8x - 2y = -14
-8x - 80y = -232

This step is like adding a special ingredient to our recipe – it might seem like a small change, but it sets the stage for the magic to happen in the next step.

Step 3: Add the Equations

Now comes the fun part! We add the two equations together. Notice what happens when we add the x terms:

(8x - 2y) + (-8x - 80y) = -14 + (-232)

The x terms cancel each other out (8x + (-8x) = 0), which is exactly what we wanted! This leaves us with:

-82y = -246

This step is like the grand reveal in a magic trick – the variable we wanted to disappear has vanished, leaving us with a much simpler equation to solve.

Step 4: Solve for the Remaining Variable

We now have a simple equation with just one variable, y. To solve for y, we divide both sides of the equation by -82:

-82y / -82 = -246 / -82
y = 3

So, we've found that y = 3! This is a major breakthrough – we've discovered one piece of the puzzle.

Step 5: Substitute

Now that we know the value of y, we can substitute it back into one of the original equations to solve for x. Let’s use the first original equation (8x - 2y = -14) because it looks a bit simpler:

8x - 2(3) = -14
8x - 6 = -14

Now, we add 6 to both sides:

8x = -8

Finally, we divide both sides by 8:

x = -1

So, we've found that x = -1! We've now solved for both variables – it's like finding the hidden treasure at the end of a long quest.

Step 6: Check

It's always a good idea to check our solution to make sure we didn't make any mistakes. To do this, we substitute both x = -1 and y = 3 into both original equations:

Equation 1:

8(-1) - 2(3) = -14
-8 - 6 = -14
-14 = -14  (Correct!)

Equation 2:

-(-1) - 10(3) = -29
1 - 30 = -29
-29 = -29  (Correct!)

Our solution checks out in both equations! This gives us confidence that we've found the correct answer.

The Solution

Therefore, the solution to the system of equations is x = -1 and y = 3. We can write this as an ordered pair: (-1, 3). This means that the point (-1, 3) is the intersection of the two lines represented by our equations. It’s like finding the exact spot on a map where two roads meet.

Tips and Tricks for Using Elimination

Here are a few extra tips and tricks to keep in mind when using the elimination method:

  • Choose Wisely: When deciding which variable to eliminate, look for coefficients that are either opposites or easy multiples of each other. This can save you steps.
  • Multiply Carefully: Make sure you multiply every term in the equation by the constant, not just the terms with the variable you're trying to eliminate.
  • Watch the Signs: Pay close attention to the signs (positive and negative) when adding the equations. A small mistake with a sign can throw off your entire solution.
  • Don't Be Afraid to Multiply Both Equations: Sometimes, you might need to multiply both equations by different constants to get the coefficients to match up. This is perfectly okay!
  • Check, Check, Check: Always check your solution by substituting the values back into the original equations. This is the best way to catch any errors.

Common Mistakes to Avoid

Even with a clear method, it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Forgetting to Multiply All Terms: When multiplying an equation by a constant, make sure you multiply every term on both sides of the equation. Leaving out a term is a common mistake that can lead to an incorrect solution.
  • Sign Errors: Be extra careful with your signs, especially when adding the equations. A simple sign error can completely change the outcome.
  • Incorrect Substitution: When substituting the value of one variable into an equation, make sure you substitute it correctly. Double-check your work to avoid errors.
  • Skipping the Check: It's tempting to skip the check step, but it's crucial for catching mistakes. Always take the time to substitute your solution back into the original equations.

Practice Makes Perfect

The best way to master the elimination method is to practice! Try solving different systems of linear equations with varying levels of complexity. The more you practice, the more comfortable and confident you'll become. Think of it like learning a new dance – the more you practice the steps, the smoother your moves will become.

Conclusion

And there you have it! We've successfully solved a system of linear equations using the elimination method. Remember, the key is to align the equations, manipulate them to eliminate a variable, solve for the remaining variable, substitute back, and always check your work. With practice, you'll become a pro at solving systems of equations using elimination. So go ahead, give it a try, and happy solving!