Solving 2x+y=2 And X+5y=1 With Elimination Method A Step-by-Step Guide

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Hey guys! Ever found yourself staring at a system of equations, feeling like you're trying to decipher an ancient code? Don't worry, you're not alone! Many students find solving systems of equations a bit tricky at first, but with the right method, it becomes a breeze. Today, we're going to dive deep into the elimination method, a powerful tool for solving these mathematical puzzles. We'll use the example of the system 2x + y = 2 and x + 5y = 1 to illustrate each step. So, buckle up and let's get started!

Understanding Systems of Equations

Before we jump into the elimination method, let's quickly recap what a system of equations actually is. Basically, it's a set of two or more equations that share the same variables. Our goal is to find the values of these variables that satisfy all the equations in the system simultaneously. Think of it like finding the perfect key that unlocks all the locks in a set. In our example, we have two equations:

  1. 2x + y = 2
  2. x + 5y = 1

We need to find the values of x and y that make both of these equations true. There are several methods to solve systems of equations, such as substitution, graphing, and, of course, elimination. Today, we're focusing on elimination because it's particularly efficient when the coefficients of one of the variables are easily made opposites.

The Elimination Method: A Detailed Walkthrough

The elimination method, as the name suggests, involves eliminating one of the variables by manipulating the equations. The core idea is to add or subtract the equations in such a way that one variable cancels out, leaving us with a single equation in one variable. This single equation is then easy to solve. Let's break down the steps involved in solving our system using the elimination method:

Step 1: Prepare the Equations

The first step is to make sure the equations are lined up nicely, with the x terms, y terms, and constants in their respective columns. This makes it easier to see which variable we can eliminate. Our equations are already in this format:

  1. 2x + y = 2
  2. x + 5y = 1

Step 2: Choose a Variable to Eliminate

Now comes the strategic part: choosing which variable to eliminate. We want to pick the variable that's easiest to get rid of. Looking at our equations, we can see that the coefficient of x in the first equation is 2, and in the second equation, it's 1. Similarly, the coefficient of y in the first equation is 1, and in the second equation, it's 5. It seems easier to eliminate x because we can easily multiply the second equation by -2 to make the x coefficients opposites.

Step 3: Multiply One or Both Equations

This is where the magic happens! We need to multiply one or both equations by a constant so that the coefficients of the variable we want to eliminate are opposites. As we discussed, we'll multiply the second equation by -2:

-2 * (x + 5y) = -2 * 1

This gives us:

-2x - 10y = -2

Now we have a modified system:

  1. 2x + y = 2
  2. -2x - 10y = -2

Notice that the coefficients of x are now 2 and -2, which are opposites!

Step 4: Add the Equations

Now comes the elimination part! We add the two equations together, term by term:

(2x + y) + (-2x - 10y) = 2 + (-2)

This simplifies to:

-9y = 0

Look at that! The x terms have completely canceled out, leaving us with a simple equation in y.

Step 5: Solve for the Remaining Variable

Solving for y is now straightforward. We divide both sides of the equation by -9:

y = 0 / -9

y = 0

So, we've found the value of y! It's 0.

Step 6: Substitute to Find the Other Variable

Now that we know y = 0, we can substitute this value into either of the original equations to solve for x. Let's use the first equation:

2x + y = 2

Substitute y = 0:

2x + 0 = 2

2x = 2

Divide both sides by 2:

x = 1

So, we've found the value of x! It's 1.

Step 7: Check Your Solution

It's always a good idea to check your solution to make sure it's correct. We do this by substituting the values of x and y into both of the original equations. If both equations are true, then our solution is correct.

Let's check with the first equation:

2x + y = 2

Substitute x = 1 and y = 0:

2(1) + 0 = 2

2 = 2 (This is true!)

Now let's check with the second equation:

x + 5y = 1

Substitute x = 1 and y = 0:

1 + 5(0) = 1

1 = 1 (This is also true!)

Since our solution satisfies both equations, we know it's correct.

The Solution

Therefore, the solution to the system of equations 2x + y = 2 and x + 5y = 1 is x = 1 and y = 0. We can write this as an ordered pair: (1, 0).

Why the Elimination Method Works: A Deeper Dive

Now that we've successfully solved the system, let's take a moment to understand why the elimination method works. The key is that we're performing operations that don't change the solution set of the system. Multiplying an equation by a constant simply scales the equation, but it doesn't change the line it represents. Adding two equations together creates a new equation that represents a combination of the original two lines. The point where the original lines intersect is the solution to the system, and this point will also lie on the new line created by adding the equations.

By strategically choosing our multipliers, we can create opposite coefficients for one of the variables. When we add the equations, this variable cancels out, leaving us with an equation in a single variable. This is a clever way to simplify the problem and isolate one variable at a time.

Tips and Tricks for Mastering Elimination

  • Choose wisely: When deciding which variable to eliminate, look for the one that will require the least amount of multiplication. Sometimes, you only need to multiply one equation, while other times, you might need to multiply both.
  • Pay attention to signs: Be careful with negative signs! A small mistake can throw off your entire solution. Double-check your work, especially when multiplying by negative numbers.
  • Don't be afraid to multiply both equations: If neither variable has coefficients that are easy to make opposites, you might need to multiply both equations by different constants. This is perfectly fine!
  • Practice, practice, practice: The more you practice the elimination method, the more comfortable you'll become with it. Try solving different systems of equations with varying levels of difficulty.

Real-World Applications of Systems of Equations

You might be wondering,