Solving X+3y=3 And X-5y=-5 By Equalization Method A Step-by-Step Guide

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Hey guys! Ever stumbled upon a system of equations and felt like you're in a mathematical maze? Don't worry, we've all been there! Today, we're going to break down a super handy method called the equalization method to solve a system of linear equations. We'll use the example of solving x + 3y = 3 and x - 5y = -5. Trust me, by the end of this guide, you'll be solving these like a pro. So, grab your pencils and let's dive in!

What is the Equalization Method?

The equalization method is a technique used to solve systems of equations, particularly when you have two equations with two variables. The basic idea is to isolate the same variable in both equations and then set the resulting expressions equal to each other. This eliminates one variable, leaving you with a single equation in one variable, which you can easily solve. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. Think of it as a mathematical shortcut to unravel the mystery of those pesky x and y values!

Why Choose the Equalization Method?

Okay, so you might be wondering, why should I bother learning this method when there are others out there? Well, the equalization method is particularly useful when the equations are already set up in a way that makes isolating a variable relatively easy. It's also a great method for building a solid understanding of how systems of equations work. By mastering this method, you'll have another powerful tool in your mathematical toolkit, making you a more versatile problem-solver. Plus, it's kinda cool to see how you can manipulate equations to reveal the hidden solutions!

Step-by-Step Guide to Solving x+3y=3 and x-5y=-5

Alright, let’s get down to business and solve the system of equations x + 3y = 3 and x - 5y = -5 using the equalization method. I’ll walk you through each step, so you can follow along and see exactly how it's done. No more feeling lost in a sea of numbers and variables – we’re going to conquer this together!

Step 1: Isolate One Variable in Both Equations

The first crucial step in the equalization method is to isolate the same variable in both equations. In our case, let’s isolate x because it looks simpler to deal with. This means we want to rewrite each equation so that x is by itself on one side.

Equation 1: x + 3y = 3

To isolate x, we need to get rid of the 3y term. We can do this by subtracting 3y from both sides of the equation:

x + 3y - 3y = 3 - 3y

This simplifies to:

x = 3 - 3y

So, we've successfully isolated x in the first equation.

Equation 2: x - 5y = -5

Now, let's do the same for the second equation. To isolate x, we need to get rid of the -5y term. We can do this by adding 5y to both sides of the equation:

x - 5y + 5y = -5 + 5y

This simplifies to:

x = -5 + 5y

Great! We’ve isolated x in both equations. Now we’re ready for the next step.

Step 2: Set the Expressions Equal to Each Other

Now that we have x isolated in both equations, we have two expressions that are both equal to x. This means we can set these expressions equal to each other. Remember, we found:

  • x = 3 - 3y
  • x = -5 + 5y

Since both of these are equal to x, we can write:

3 - 3y = -5 + 5y

This is the heart of the equalization method! We've eliminated x and now we have a single equation with just one variable, y. The math gods are smiling upon us!

Step 3: Solve for the Remaining Variable (y)

Alright, let's solve for y in the equation we just got: 3 - 3y = -5 + 5y. Our goal is to get all the y terms on one side and all the constant terms on the other side.

First, let's add 3y to both sides of the equation to get rid of the -3y on the left:

3 - 3y + 3y = -5 + 5y + 3y

This simplifies to:

3 = -5 + 8y

Next, let's add 5 to both sides of the equation to get rid of the -5 on the right:

3 + 5 = -5 + 5 + 8y

This simplifies to:

8 = 8y

Finally, to solve for y, we need to divide both sides of the equation by 8:

8 / 8 = 8y / 8

This gives us:

1 = y

So, we've found that y = 1! We’re halfway there – awesome job!

Step 4: Substitute the Value Back into One of the Equations to Solve for x

Now that we know y = 1, we can substitute this value back into either of our original equations to find the value of x. It doesn't matter which equation we choose, we should get the same answer. Let's use the first equation we derived, x = 3 - 3y, because it looks a little simpler.

Substitute y = 1 into x = 3 - 3y:

x = 3 - 3(1)

x = 3 - 3

x = 0

So, we've found that x = 0! We’ve solved for both x and y – high five!

Step 5: Check Your Solution

Before we celebrate too much, it’s always a good idea to check our solution to make sure we didn’t make any sneaky errors. We can do this by plugging the values we found for x and y back into both of the original equations.

Original Equation 1: x + 3y = 3

Substitute x = 0 and y = 1:

0 + 3(1) = 3

0 + 3 = 3

3 = 3

This checks out!

Original Equation 2: x - 5y = -5

Substitute x = 0 and y = 1:

0 - 5(1) = -5

0 - 5 = -5

-5 = -5

This also checks out! Since our solution satisfies both equations, we can be confident that we’ve solved the system correctly. Woot woot!

Conclusion

And there you have it! We've successfully solved the system of equations x + 3y = 3 and x - 5y = -5 using the equalization method. We isolated x in both equations, set the expressions equal to each other, solved for y, substituted the value of y back to find x, and then checked our solution. Not too shabby, right?

The equalization method might seem a bit tricky at first, but with a little practice, you'll become a master at it. Remember, the key is to break down the problem into manageable steps and take it one step at a time. So, next time you encounter a system of equations, don't sweat it – just remember the equalization method and you'll be well on your way to solving it. Keep practicing, and you'll be acing those math problems in no time! You got this!

Practice Problems

Want to put your newfound skills to the test? Here are a couple of practice problems for you to try. Remember to follow the steps we discussed, and don’t be afraid to double-check your work. Happy solving!

  1. Solve the system: 2x + y = 7 and x - y = 2
  2. Solve the system: 3x - 2y = 5 and x + y = 5

Further Learning

If you're hungry for more math knowledge, there are tons of resources out there to help you expand your understanding of systems of equations and other mathematical concepts. Check out online tutorials, math textbooks, and educational websites. The more you explore, the more confident you'll become in your math abilities. Keep up the great work, and remember, math can be fun!