Solving Rational Equations A Step-by-Step Guide To 1/x + 1/(x-10) = (x-9)/(x-10)

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of equations, tackling a specific problem that might seem a bit daunting at first glance. But don't worry, we'll break it down step by step, making sure everyone, from math newbies to seasoned pros, can follow along. Our mission? To solve the equation $\frac{1}{x}+\frac{1}{x-10}=\frac{x-9}{x-10}$ and uncover its solutions, presenting them in the correct order. So, grab your calculators, put on your thinking caps, and let's get started!

Understanding the Equation

Before we jump into the solution, let's take a moment to understand the equation we're dealing with. We have a rational equation, which means it involves fractions with variables in the denominator. These types of equations require a bit of extra care because we need to avoid values of x that would make the denominator zero, as division by zero is undefined.

The equation in question is $\frac{1}{x}+\frac{1}{x-10}=\frac{x-9}{x-10}$. Notice that we have x and x-10 in the denominators. This tells us that x cannot be 0 and x cannot be 10, because these values would make the denominators zero. Keeping these restrictions in mind is crucial as we solve the equation.

Why are these restrictions important, you ask? Well, if we were to find a solution that makes the denominator zero, that solution would be extraneous. An extraneous solution is a value that we obtain through the solving process, but it doesn't actually satisfy the original equation. It's like a false lead in a detective novel – it looks promising, but it ultimately leads nowhere. So, let's keep those restrictions – x ≠ 0 and x ≠ 10 – at the forefront of our minds as we proceed.

The Step-by-Step Solution

Now, let's roll up our sleeves and get to the heart of the matter: solving the equation. We'll take a systematic approach, breaking down the process into manageable steps.

Step 1: Eliminate the Fractions

The first thing we want to do is get rid of those pesky fractions. To do this, we'll multiply both sides of the equation by the least common denominator (LCD). The LCD is the smallest expression that all the denominators divide into evenly. In our case, the denominators are x and x-10, so the LCD is simply their product: x(x-10).

Multiplying both sides of the equation by x(x-10), we get:

$$x(x-10)\left(\frac{1}{x}+\frac{1}{x-10}\right)=x(x-10)\left(\frac{x-9}{x-10}\right)$$

Distributing x(x-10) on both sides, we have:

$$x(x-10)\cdot\frac{1}{x}+x(x-10)\cdot\frac{1}{x-10}=x(x-10)\cdot\frac{x-9}{x-10}$$

Now, we can cancel out common factors:

$$(x-10)+x=x(x-9)$$

See how the fractions have vanished? We're making progress!

Step 2: Simplify and Rearrange

Next, let's simplify the equation by combining like terms and expanding any products. On the left side, we have (x-10) + x, which simplifies to 2x - 10. On the right side, we have x(x-9), which expands to x^2 - 9x. So, our equation now looks like this:

$$2x-10=x^2-9x$$

To solve this quadratic equation, we need to get all the terms on one side, setting the equation equal to zero. Let's subtract 2x from both sides and add 10 to both sides:

$$0=x^2-9x-2x+10$$

Combining like terms, we get:

$$0=x^2-11x+10$$

Now we have a standard quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = -11, and c = 10.

Step 3: 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, factoring seems like the most straightforward approach. We're looking for two numbers that multiply to 10 (the constant term) and add up to -11 (the coefficient of the x term). Those numbers are -1 and -10. So, we can factor the quadratic as:

$$0=(x-1)(x-10)$$

To find the solutions, we set each factor equal to zero:

$$x-1=0 \text{ or } x-10=0$$

Solving for x, we get:

$$x=1 \text{ or } x=10$$

Step 4: Check for Extraneous Solutions

Remember those restrictions we talked about earlier? We need to check if our solutions, x = 1 and x = 10, satisfy those restrictions. We know that x cannot be 0 and x cannot be 10. Uh-oh! One of our solutions, x = 10, is an extraneous solution because it makes the denominator x-10 zero.

This means that x = 10 is not a valid solution to the original equation. We discard it.

Step 5: State the Solutions

The only valid solution we have is x = 1. Since the question asks for two solutions, and we only have one, we enter "n.a." for the second solution.

So, the solutions are:

$$x=1 \text{ or } x=\text{n.a.}$$

Alternative Methods for Solving Quadratic Equations

While we successfully solved the quadratic equation by factoring, it's worth mentioning other methods you can use. These methods come in handy when factoring isn't so obvious or when the quadratic is more complex.

The Quadratic Formula

The quadratic formula is a powerful tool that can solve any quadratic equation, regardless of whether it can be factored easily. The formula is:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In our case, a = 1, b = -11, and c = 10. Plugging these values into the formula, we get:

$$x=\frac{-(-11)\pm\sqrt{(-11)^2-4(1)(10)}}{2(1)}$$

$$x=\frac{11\pm\sqrt{121-40}}{2}$$

$$x=\frac{11\pm\sqrt{81}}{2}$$

$$x=\frac{11\pm9}{2}$$

This gives us two possible solutions:

$$x=\frac{11+9}{2}=\frac{20}{2}=10$$

$$x=\frac{11-9}{2}=\frac{2}{2}=1$$

As we found before, x = 10 is an extraneous solution, and x = 1 is the only valid solution.

Completing the Square

Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side. While it's a bit more involved than factoring or using the quadratic formula, it's a valuable technique to have in your mathematical toolkit.

To complete the square for our equation x^2 - 11x + 10 = 0, we first move the constant term to the right side:

$$x^2-11x=-10$$

Next, we take half of the coefficient of the x term (which is -11), square it ((-11/2)^2 = 121/4), and add it to both sides:

$$x^2-11x+\frac{121}{4}=-10+\frac{121}{4}$$

Now, the left side is a perfect square trinomial:

$$\left(x-\frac{11}{2}\right)^2=\frac{81}{4}$$

Taking the square root of both sides, we get:

$$x-\frac{11}{2}=\pm\frac{9}{2}$$

Solving for x, we have:

$$x=\frac{11}{2}\pm\frac{9}{2}$$

This gives us two possible solutions:

$$x=\frac{11+9}{2}=10$$

$$x=\frac{11-9}{2}=1$$

Again, we find that x = 10 is extraneous, and x = 1 is the only valid solution.

Common Pitfalls and How to Avoid Them

Solving equations, especially rational equations and quadratic equations, can be tricky. There are a few common pitfalls that students often encounter. Let's discuss these pitfalls and how to avoid them.

Forgetting to Check for Extraneous Solutions

As we've seen, extraneous solutions can sneak into our calculations, especially when dealing with rational equations. It's crucial to always check your solutions against the original equation to make sure they don't make any denominators zero. Make it a habit to identify the restrictions on the variable before you start solving the equation. This will serve as a reminder to check your answers later.

Making Sign Errors

Sign errors are easy to make, especially when dealing with negative numbers. Pay close attention to signs when distributing, combining like terms, and applying the quadratic formula. A small sign error can throw off the entire solution. Double-check your work, and if possible, use a calculator to verify your calculations.

Incorrectly Applying the Order of Operations

The order of operations (PEMDAS/BODMAS) is the bedrock of mathematical calculations. Make sure you're following the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). A mistake in the order of operations can lead to an incorrect result.

Difficulty Factoring Quadratic Equations

Factoring quadratics can be challenging, especially when the coefficients are large or the quadratic is not easily factorable. If you're struggling with factoring, remember the quadratic formula is your trusty backup. It can solve any quadratic equation, regardless of its factorability. Also, practice makes perfect! The more you factor, the better you'll become at recognizing patterns and applying factoring techniques.

Not Understanding the Concepts

Perhaps the biggest pitfall is trying to solve problems without truly understanding the underlying concepts. Math is not just about memorizing formulas and procedures; it's about understanding why those formulas and procedures work. If you find yourself struggling, take the time to review the concepts, seek help from your teacher or a tutor, and work through plenty of examples. A solid conceptual understanding will make problem-solving much easier and more rewarding.

Real-World Applications of Equations

You might be wondering, "When am I ever going to use this in real life?" Well, equations are not just abstract mathematical concepts; they're powerful tools that help us understand and solve problems in the real world. From engineering to finance to physics, equations are used to model and analyze a wide range of phenomena.

Engineering

Engineers use equations to design bridges, buildings, and machines. They need to calculate stresses, strains, and forces to ensure that their designs are safe and efficient. Quadratic equations, in particular, are used in many engineering calculations, such as determining the trajectory of a projectile or the shape of a suspension cable.

Finance

In finance, equations are used to calculate interest rates, loan payments, and investment returns. Financial analysts use mathematical models to predict market trends and assess risk. Understanding equations is essential for making informed financial decisions.

Physics

Physics is heavily reliant on equations to describe the laws of nature. From Newton's laws of motion to Einstein's theory of relativity, equations are used to model everything from the motion of planets to the behavior of subatomic particles. Solving equations is fundamental to understanding the physical world.

Computer Science

Equations are also used in computer science for tasks such as algorithm design, data analysis, and machine learning. Computer scientists use mathematical models to develop software and analyze data. Understanding equations is key to becoming a proficient computer scientist.

Everyday Life

Even in our everyday lives, we use equations without even realizing it. For example, when we calculate how much time it will take to drive somewhere, or how much money we'll save with a discount, we're using equations. The ability to solve equations is a valuable skill that can help us make better decisions in all aspects of our lives.

Conclusion: Mastering Equations for Mathematical Success

Congratulations, you've made it to the end of our comprehensive guide to solving the equation $\frac{1}{x}+\frac{1}{x-10}=\frac{x-9}{x-10}$. We've journeyed through the step-by-step solution, explored alternative methods, discussed common pitfalls, and even touched on real-world applications. Hopefully, you now have a solid understanding of how to tackle this type of problem and a deeper appreciation for the power of equations.

Remember, the key to mastering mathematics is practice. Work through as many problems as you can, and don't be afraid to make mistakes. Mistakes are learning opportunities! Keep honing your skills, and you'll be well on your way to mathematical success.

So, go forth and conquer those equations! You've got this!