Elimination Method: Step-by-Step Solution

Let's dive into the elimination method for solving systems of equations. This method is a powerful tool for finding the values of unknown variables when you have multiple equations. We'll walk through a step-by-step example to make it crystal clear. Get ready to master this technique!

Understanding the Elimination Method

The elimination method, also known as the addition method, works by manipulating the equations in a system so that when you add them together, one of the variables cancels out. This leaves you with a single equation with a single variable, which is easy to solve. 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.

The core idea behind the elimination method is to strategically multiply one or both equations by constants so that the coefficients of one variable are opposites. For example, if one equation has 2x and the other has -2x, adding the equations will eliminate x. If the coefficients are not direct opposites, we can manipulate the equations to create that situation. This usually involves multiplying one or both equations by a carefully chosen number.

Let's break down the general steps involved:

1. Align the Equations: Make sure the like terms (x-terms, y-terms, constants) are lined up in columns.
2. Multiply (if necessary): Multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 3y and -3y).
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 Your Solution: Plug both values back into the original equations to verify they satisfy both.

By following these steps, the elimination method becomes a straightforward process. Each step builds upon the previous, ultimately leading to the solution. It's like following a recipe – each ingredient (or step) is essential for the final outcome. So, let's put this into practice with a specific example.

Example: Solving a System of Equations

Consider the following system of equations:

3x + 2y = 22
-x + 4y = 2

Our goal is to find the values of x and y that satisfy both equations simultaneously. Let's apply the elimination method step-by-step.

Step 1: Align the Equations

In this case, the equations are already nicely aligned. The x terms, y terms, and constants are lined up in columns, which makes the next steps easier to manage. Proper alignment is crucial because it allows us to directly add or subtract terms without confusion.

Step 2: Multiply (if necessary)

We need to make the coefficients of either x or y opposites. Notice that the coefficient of x in the first equation is 3, and in the second equation, it's -1. If we multiply the second equation by 3, the coefficient of x will become -3, which is the opposite of 3. This sets us up perfectly for eliminating x. So, let’s multiply the entire second equation by 3:

3 * (-x + 4y) = 3 * 2
-3x + 12y = 6

Now we have a modified second equation that will help us eliminate the x variable. The decision to multiply the second equation by 3 was strategic; it created the necessary opposite coefficients for x, setting the stage for the next step.

Step 3: Add the Equations

Now, let's add the first equation to the modified second equation:

  3x + 2y = 22
+ (-3x + 12y = 6)
----------------
  0x + 14y = 28

Notice how the x terms cancel out (3x and -3x), leaving us with an equation in terms of y only. This is the key step in the elimination method. We’ve successfully eliminated one variable, simplifying the problem significantly. The resulting equation, 14y = 28, is much easier to solve.

Step 4: Solve for the Remaining Variable

We now have the equation 14y = 28. To solve for y, we divide both sides by 14:

14y / 14 = 28 / 14
y = 2

So, we've found that y = 2. This is a major step forward. Knowing the value of y allows us to substitute it back into one of the original equations to find x. We're halfway to solving the system!

Step 5: Substitute

Substitute the value of y (which is 2) into either of the original equations. Let's use the first equation:

3x + 2y = 22
3x + 2(2) = 22
3x + 4 = 22

Now, we solve for x:

3x = 22 - 4
3x = 18
x = 18 / 3
x = 6

So, we've found that x = 6. Now we have both x and y values, but it’s crucial to verify that these values satisfy both original equations. This ensures we haven't made any errors along the way.

Step 6: Check Your Solution

Plug x = 6 and y = 2 into both original equations to check:

• Equation 1: 3x + 2y = 22

  3(6) + 2(2) = 18 + 4 = 22` (Correct!)
  

• Equation 2: -x + 4y = 2

  -6 + 4(2) = -6 + 8 = 2` (Correct!)
  

Since both equations are satisfied, our solution is correct. We have successfully found the values of x and y that make both equations true.

The Solution as an Ordered Pair

The solution to the system of equations is x = 6 and y = 2. We write this as an ordered pair: (6, 2). This ordered pair represents the point where the two lines represented by the equations intersect on a graph.

Why the Elimination Method Works

The elimination method works because adding equal quantities to equal quantities preserves equality. When we multiply an equation by a constant, we're essentially multiplying both sides by the same number, so the equation remains balanced. Similarly, when we add two equations together, we're adding the same quantity to both sides of the combined equation, maintaining the balance. The strategic multiplication and addition steps are designed to eliminate one variable, leading us to a simpler equation that we can solve.

Think of it like a balancing scale. If you add the same weight to both sides, the scale stays balanced. The same principle applies to equations. We manipulate them while maintaining the equality, ultimately isolating the variables we want to solve for.

When to Use the Elimination Method

The elimination method is particularly useful when the coefficients of one variable in the two equations are either the same or easily made the same (or opposites) by multiplication. It's a straightforward and efficient method when dealing with systems where variables can be readily eliminated through addition or subtraction. However, if the equations are already solved for one variable (e.g., y = ...), the substitution method might be more convenient.

Consider these scenarios:

• Coefficients are opposites: If you see coefficients like 2x and -2x, the elimination method is a natural choice.
• Easy to create opposites: If you can easily multiply one equation to create opposite coefficients (as we did in our example), this method is highly effective.
• Standard form equations: When equations are in standard form (Ax + By = C), the elimination method often flows smoothly.

By recognizing these patterns, you can quickly assess whether the elimination method is the best approach for a given system of equations.

Tips and Tricks for Success

• Always double-check your work: Mistakes can easily happen during multiplication and addition. Taking a moment to review each step will save you time in the long run.
• Be careful with signs: Pay close attention to positive and negative signs, especially when multiplying and adding equations. A simple sign error can throw off the entire solution.
• Stay organized: Keep your work neat and organized. Write the equations clearly and align the terms properly. This will help you avoid mistakes and make it easier to follow your steps.
• Practice, practice, practice: The more you practice, the more comfortable you'll become with the elimination method. Work through various examples to build your skills and confidence.

By following these tips, you'll enhance your accuracy and efficiency when using the elimination method to solve systems of equations.

Conclusion

The elimination method is a powerful and versatile technique for solving systems of equations. By strategically manipulating equations to eliminate one variable, we can simplify the problem and find the values of the unknowns. Remember the steps: align, multiply, add, solve, substitute, and check. With practice, you'll master this method and be able to confidently solve a wide range of systems of equations. The solution to our example system is the ordered pair (6, 2). Keep practicing, and you'll become a pro at solving systems of equations!

For further learning and more examples, you might find helpful resources at Khan Academy's Systems of Equations. This website provides comprehensive lessons and practice exercises to solidify your understanding.