Solving Simultaneous Equations: Finding Max X And Corresponding Y

Hey guys! Today, we're diving into a fun math problem that involves solving simultaneous equations. Specifically, we're going to find the largest value of x that satisfies two equations and then determine the corresponding value of y. Sounds like a puzzle, right? Let's get started!

Understanding the Problem

Before we jump into solving, let's break down the problem. We have two equations:

1. x + y = 2
2. 3x² - 2x + y² = 2

Our goal is to find the largest value of x that makes both of these equations true at the same time. Once we find that x, we'll also find the y that goes with it. This is a classic problem in algebra, and it's a great way to flex those equation-solving muscles.

Step-by-Step Solution

Okay, let's get down to business and solve this step by step.

Step 1: Express y in Terms of x

The first equation, x + y = 2, is pretty simple. We can easily rearrange it to express y in terms of x. This means we want to get y all by itself on one side of the equation. So, we subtract x from both sides:

y = 2 - x

Now we have an expression for y that we can use in the second equation. This is a crucial step because it allows us to reduce the problem from two variables (x and y) to just one variable (x).

Step 2: Substitute into the Second Equation

Next, we're going to substitute our expression for y (which is 2 - x) into the second equation, 3x² - 2x + y² = 2. This might look a little intimidating, but don't worry, we'll take it slow.

Replace y with (2 - x) in the second equation:

3x² - 2x + (2 - x)² = 2

Now we have an equation that only involves x. This is great news because we can now solve for x.

Step 3: Expand and Simplify

Before we can solve for x, we need to simplify the equation. This means expanding the squared term (2 - x)² and then combining like terms. Remember, (2 - x)² means (2 - x) multiplied by itself.

Let's expand (2 - x)²:

(2 - x)² = (2 - x)(2 - x) = 4 - 2x - 2x + x² = 4 - 4x + x²

Now, substitute this back into our equation:

3x² - 2x + (4 - 4x + x²) = 2

Combine like terms:

3x² + x² - 2x - 4x + 4 = 2

4x² - 6x + 4 = 2

Step 4: Rearrange into a Quadratic Equation

To solve for x, we need to rearrange the equation into the standard quadratic form, which is ax² + bx + c = 0. To do this, we subtract 2 from both sides:

4x² - 6x + 4 - 2 = 0

4x² - 6x + 2 = 0

Now we have a quadratic equation in standard form. Time to solve it!

Step 5: Solve the Quadratic Equation

There are a few ways to solve a quadratic equation: factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula, which is:

x = (-b ± √(b² - 4ac)) / (2a)

In our equation, 4x² - 6x + 2 = 0, we have a = 4, b = -6, and c = 2. Let's plug these values into the quadratic formula:

x = (-(-6) ± √((-6)² - 4 * 4 * 2)) / (2 * 4)

x = (6 ± √(36 - 32)) / 8

x = (6 ± √4) / 8

x = (6 ± 2) / 8

This gives us two possible values for x:

x₁ = (6 + 2) / 8 = 8 / 8 = 1

x₂ = (6 - 2) / 8 = 4 / 8 = 1/2

So, we have two solutions for x: 1 and 1/2. But remember, we're looking for the largest value of x. So, the largest value of x that satisfies the equations is 1.

Step 6: Find the Corresponding Value of y

Now that we've found the largest x, we need to find the corresponding y. We can use our earlier expression for y in terms of x: y = 2 - x. Plug in x = 1:

y = 2 - 1 = 1

So, when x is 1, y is also 1.

The Solution

We've done it! We found the largest value of x that satisfies the simultaneous equations, and we found the corresponding value of y. Here's our solution:

The largest value of x is 1, and the corresponding value of y is 1.

Checking Our Work

It's always a good idea to check our work to make sure we didn't make any mistakes. Let's plug x = 1 and y = 1 back into our original equations:

1. x + y = 1 + 1 = 2 (Correct!)
2. 3x² - 2x + y² = 3(1)² - 2(1) + (1)² = 3 - 2 + 1 = 2 (Correct!)

Both equations are satisfied, so we know our solution is correct. High five!

Why This Matters

Solving simultaneous equations is a fundamental skill in mathematics. It comes up in all sorts of real-world applications, from engineering to economics to computer science. Understanding how to solve these types of problems can help you model and analyze complex systems.

Tips for Solving Simultaneous Equations

Here are a few tips that might help you when you're tackling similar problems:

• Isolate a Variable: Look for the easiest equation to isolate one of the variables. This makes substitution much simpler.
• Simplify: Always simplify your equations as much as possible before solving. This reduces the chance of making mistakes.
• Check Your Answers: Plugging your solutions back into the original equations is a crucial step to ensure accuracy.

Conclusion

So, there you have it! We successfully found the largest value of x and the corresponding y that satisfy the given simultaneous equations. Remember, practice makes perfect, so keep working on these types of problems. You'll become a pro in no time!

Solving simultaneous equations might seem tricky at first, but with practice, you'll get the hang of it. The key is to break the problem down into smaller steps, stay organized, and double-check your work. Math can be fun, especially when you conquer a challenging problem. Keep exploring, keep learning, and you'll be amazed at what you can achieve! Keep your mind sharp, and who knows, maybe you'll discover new ways to solve equations that even mathematicians haven't thought of yet!