Solving Linear Equations: Find X & Y Values!

by Chloe Fitzgerald 45 views

Hey guys! Today, we're diving into the world of linear equations. Solving these equations might seem daunting at first, but trust me, it's like piecing together a puzzle. We're going to tackle a system of two linear equations, find the values of X and Y that fit perfectly, and make the whole process super clear. So, buckle up, grab your math hats, and let's get started!

The Challenge: Cracking the Code of Linear Equations

Our mission, should we choose to accept it (and we totally do!), is to solve the following system of equations:

  1. X - 2y = 14
  2. 2x + y = -22

We need to find the magic numbers for X and Y that make both of these equations true. Think of it like a secret code where X and Y are the unknown symbols. We've got a few options to choose from:

  • A) X = 10, Y = -8
  • B) X = 12, Y = -5
  • C) X = 8, Y = -3
  • D) X = 6, Y = -1

So, how do we crack this code? Let's explore some strategies, shall we?

Method 1: The Substitution Game

The substitution method is like a strategic swap in a game. We'll isolate one variable in one equation and then substitute its value into the other equation. This way, we'll get an equation with just one variable, making it much easier to solve. Let's break it down:

Step 1: Isolate a Variable

Let's take the first equation, X - 2y = 14, and isolate X. To do this, we'll add 2y to both sides:

X = 14 + 2y

Now, we have X all by itself, which is exactly what we want.

Step 2: Substitute and Simplify

Now comes the fun part – substitution! We'll take the value of X (which is 14 + 2y) and plug it into the second equation, 2x + y = -22:

2(14 + 2y) + y = -22

See what we did there? We replaced X with its equivalent expression. Now, let's simplify this equation. First, distribute the 2:

28 + 4y + y = -22

Combine those 'y' terms:

28 + 5y = -22

Step 3: Solve for Y

Now we've got a simple equation with just Y. Let's isolate Y by subtracting 28 from both sides:

5y = -22 - 28

5y = -50

Finally, divide both sides by 5 to get Y:

y = -10

We've found our Y! It's -10. But we're not done yet; we still need to find X.

Step 4: Find X

We'll use the value of Y that we just found and plug it back into the equation where we isolated X: X = 14 + 2y

X = 14 + 2(-10)

X = 14 - 20

X = -6

So, X is -6. We've cracked the code using substitution!

Method 2: The Elimination Expedition

The elimination method is another cool technique. It's like a strategic subtraction game. We'll manipulate the equations so that when we add or subtract them, one of the variables disappears. Let's see how it works:

Step 1: Line 'Em Up

First, let's rewrite our equations so that the X and Y terms are lined up nicely:

  1. X - 2y = 14
  2. 2x + y = -22

Step 2: Make the Coefficients Match (or Be Opposites)

We want either the X coefficients or the Y coefficients to be the same number (but possibly with opposite signs). Let's focus on the Y coefficients. We have -2y in the first equation and +y in the second. If we multiply the second equation by 2, we'll get 2y, which is the opposite of -2y.

So, multiply equation 2 by 2:

2(2x + y) = 2(-22)

4x + 2y = -44

Now our equations look like this:

  1. X - 2y = 14
  2. 4x + 2y = -44

Step 3: Eliminate a Variable

Now we can add the two equations together. Notice what happens to the Y terms:

(X - 2y) + (4x + 2y) = 14 + (-44)

X + 4x - 2y + 2y = -30

5x = -30

The Y terms canceled out! We've eliminated Y and are left with an equation with just X.

Step 4: Solve for X

Divide both sides by 5 to get X:

X = -6

We found X! It's -6, just like before.

Step 5: Find Y

We'll plug the value of X back into either of the original equations to solve for Y. Let's use the first equation, X - 2y = 14:

-6 - 2y = 14

Add 6 to both sides:

-2y = 20

Divide both sides by -2:

y = -10

And there we have it! Y is -10, which matches our result from the substitution method.

The Verdict: Decoding the Solution

Both the substitution method and the elimination method led us to the same answer:

X = -6 Y = -10

None of the options provided (A, B, C, or D) match our solution. So, it seems like there might be a slight error in the given alternatives. But hey, that's okay! We've learned the process, and that's what truly matters.

Why This Matters: Real-World Equations

Solving linear equations isn't just about math class, guys. These skills are super useful in the real world! Think about scenarios like:

  • Budgeting: Figuring out how to spend your money wisely.
  • Cooking: Adjusting recipes to serve more or fewer people.
  • Science: Calculating distances, speeds, and other measurements.
  • Business: Analyzing costs and profits.

The ability to solve equations is like having a superpower – it helps you make informed decisions and solve problems in all areas of life.

Final Thoughts: Keep Practicing!

So, there you have it! We've conquered a system of linear equations using both substitution and elimination. Remember, practice makes perfect. The more you solve these types of problems, the more confident you'll become. Don't be afraid to try different methods and see what works best for you.

Keep those math muscles flexed, and I'll catch you in the next math adventure! Keep exploring, keep learning, and most importantly, keep having fun with math!