Solving Equations: A Step-by-Step Guide

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Solving Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a fun little algebra problem that's all about finding the values of a variable. We'll be working with a fractional expression and figuring out when it equals a specific number. Let's get started!

The Problem Unveiled

Our mission, should we choose to accept it (and we do!), is to determine the values of x for which the following expression holds true:

1x+2x+1=1\frac{1}{x} + \frac{2}{x+1} = 1

This might look a bit intimidating at first, but trust me, it's totally manageable. The key is to break it down into smaller, more digestible steps. We'll be using some fundamental algebraic techniques to isolate x and find its possible values. Are you ready to crack the code? Let's go!

This kind of problem is common in algebra, and it's a great exercise in working with fractions and equations. You'll often encounter similar problems when dealing with rates, ratios, and proportions. The skill of solving for a variable in a fractional equation is super valuable in many areas of mathematics and even in real-world applications. Being comfortable with these types of problems will boost your confidence and make you a more well-rounded problem-solver. Think of it as leveling up your math skills! We'll start by making sure we understand the expression, then methodically go through the steps to solve it. It's like a puzzle, and we have all the pieces.

First, let's just observe the equation. We have two fractions added together, and that sum is supposed to equal 1. The denominators of the fractions involve the variable x. It's important to remember that we can't have a denominator equal to zero, or things get undefined. So, we'll need to keep an eye out for values of x that would make the denominators zero, because those values aren't allowed solutions. This is an important step to keep in mind, because it will help us to eliminate any extraneous solutions that might pop up during the algebraic manipulations.

Step-by-Step Solution

Alright, buckle up, because here's how we're going to solve this equation step-by-step. I'll make sure to explain everything clearly, so even if you're new to this, you should be able to follow along. No sweat!

  1. Find a Common Denominator: The first step is to get rid of those pesky fractions. To do that, we need a common denominator. In this case, the least common denominator (LCD) is x(x + 1). Multiply both sides of the equation by this LCD.

    x(x+1)(1x+2x+1)=1β‹…x(x+1)x(x+1) \left( \frac{1}{x} + \frac{2}{x+1} \right) = 1 \cdot x(x+1)

  2. Simplify: Now, distribute x(x + 1) to each term on the left side and simplify.

    (x+1)+2x=x2+x(x+1) + 2x = x^2 + x

  3. Combine Like Terms: Combine like terms to get all the x terms together.

    3x+1=x2+x3x + 1 = x^2 + x

  4. Rearrange the Equation: Rearrange the equation to set it equal to zero. This will give us a quadratic equation.

    0=x2βˆ’2xβˆ’10 = x^2 - 2x - 1

  5. Solve the Quadratic Equation: Now, we have a quadratic equation. We can solve it using the quadratic formula:

    x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

    In our equation, a = 1, b = -2, and c = -1. Substitute these values into the formula.

    x=2Β±(βˆ’2)2βˆ’4(1)(βˆ’1)2(1)x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)}

    x=2Β±4+42x = \frac{2 \pm \sqrt{4 + 4}}{2}

    x=2Β±82x = \frac{2 \pm \sqrt{8}}{2}

    x=2Β±222x = \frac{2 \pm 2\sqrt{2}}{2}

    x=1Β±2x = 1 \pm \sqrt{2}

  6. Check for Extraneous Solutions: We need to ensure that our solutions don't make the original denominators equal to zero. We know that x cannot be 0 or -1. Since neither of our solutions, 1 + √2 and 1 - √2, are 0 or -1, they are both valid.

There you have it! The solutions to the equation are x = 1 + √2 and x = 1 - √2. Awesome job, everyone!

This process is like following a recipe, isn't it? Each step builds on the previous one, and before you know it, you've got your answer! The common denominator step is crucial because it helps us to get rid of fractions and work with a simpler equation. Combining like terms is all about tidying things up and making the equation easier to handle. Rearranging to a quadratic form allows us to use a proven method for solving for x. The quadratic formula is a super-powerful tool for solving quadratic equations, and understanding how to use it is a game-changer in algebra. And, finally, checking for extraneous solutions is a vital step because it ensures that our solutions make sense in the original equation. Math can seem abstract at times, but remember, there's always a reason for each step. Each step is built on solid mathematical principles.

Conclusion: The Grand Finale

So, to recap, we've successfully found the values of x that satisfy our original equation: x = 1 + √2 and x = 1 - √2. These are the values that make the expression $\frac{1}{x} + \frac{2}{x+1}$ equal to 1. Congratulations on working through this problem with me! You are awesome! It might seem like a lot of steps, but with practice, this will become second nature. Keep practicing, and you'll become a master of solving these kinds of equations in no time. If you got stuck at any point, don't worry! Review the steps, and try solving similar problems. That's how we learn and grow. Math is a journey, and every problem you solve is a step forward.

What we did today is a fundamental skill in algebra. The ability to solve fractional equations is the basis for a great deal of work in higher mathematics, as well as in many technical fields. By learning how to solve this equation, you have opened the door to a deeper understanding of mathematical concepts. The techniques you've learned today can be adapted to tackle a huge range of problems. So, go out there, practice, and explore the wonders of mathematics. You've got this! And remember, practice makes perfect. Keep up the great work and the effort.

Tips for Success

Want to become a super solver? Here are a few tips to help you along the way:

  • Practice Regularly: The more you practice, the better you'll become. Work through different types of problems to build your skills.
  • Understand the Concepts: Don't just memorize formulas. Make sure you understand why each step works. This will help you tackle more complex problems.
  • Break It Down: If a problem seems overwhelming, break it down into smaller steps. This makes it easier to manage.
  • Check Your Work: Always check your solutions to make sure they make sense in the original equation.
  • Don't Be Afraid to Ask: If you get stuck, don't hesitate to ask for help from a teacher, tutor, or friend. Math is easier when you work together.

By following these tips, you'll be well on your way to becoming a math whiz! Keep the momentum going! Keep learning!

Learning math should be fun and fulfilling, not something you dread! And with the correct approach and consistent practice, you'll be solving these problems like a pro! It all starts with the first step, and look at you now, tackling complex equations.

Keep up the great work, everyone! You all are amazing!