Hey there, math enthusiasts! Today, we're diving deep into the world of rational equations, and we're going to break down a problem step-by-step so you can solve them like a pro. We'll be tackling the equation:
So, grab your pencils, and let's get started!
Understanding Rational Equations
Before we jump into solving this specific equation, let's quickly recap what rational equations are. Basically, they're equations that contain fractions where the numerator and/or the denominator involve variables. In our case, we've got fractions with t + 10 in the denominator, which makes it a rational equation. The key to solving these equations lies in eliminating those pesky fractions, and that's exactly what we're going to do.
The Game Plan: Eliminating Fractions
Our main goal is to get rid of the fractions so we can work with a simpler equation. To do this, we'll use a technique called multiplying by the least common denominator (LCD). The least common denominator is the smallest expression that all the denominators in the equation divide into evenly. In our equation, we only have one denominator, which is t + 10. This makes our job a bit easier because t + 10 is also our LCD.
Step-by-Step Solution
Let's walk through the solution step by step:
-
Identify the LCD: As we discussed, the LCD for our equation is
t + 10. -
Multiply both sides of the equation by the LCD: This is the crucial step where we eliminate the fractions. We'll multiply both sides of the equation by
(t + 10):(t + 10) * [\frac{1}{t+10}+6] = (t + 10) * \frac{7}{t+10} -
Distribute and simplify: Now, we need to distribute
(t + 10)on the left side of the equation:(t + 10) * \frac{1}{t+10} + (t + 10) * 6 = (t + 10) * \frac{7}{t+10}This simplifies to:
1 + 6(t + 10) = 7Notice how the fractions are gone! We've successfully eliminated them by multiplying by the LCD. This is the power of the LCD method – it transforms a complex rational equation into a much simpler linear equation. Now, let's continue simplifying and solving for
t. -
Continue simplifying: Let's distribute the
6on the left side:1 + 6t + 60 = 7Combine the constants:
6t + 61 = 7 -
Isolate the variable: Subtract
61from both sides:6t = 7 - 616t = -54 -
Solve for t: Divide both sides by
6:t = \frac{-54}{6}t = -9
The Solution Set
So, we've found that t = -9. But before we box our answer, there's one crucial step we need to take: checking for extraneous solutions.
Checking for Extraneous Solutions
Extraneous solutions are solutions that we find algebraically, but they don't actually work in the original equation. This can happen in rational equations because we're dealing with denominators, and we need to make sure that our solution doesn't make any of the denominators equal to zero. If a solution makes a denominator zero, it's an extraneous solution, and we have to discard it.
In our case, the denominator is t + 10. If we plug in t = -9, we get -9 + 10 = 1, which is not zero. So, t = -9 is a valid solution.
Therefore, the solution set for the equation is {-9}.
Key Strategies for Solving Rational Equations
Mastering rational equations involves understanding the core principles and applying effective strategies. Here are some key techniques to keep in mind:
1. Find the Least Common Denominator (LCD)
The LCD is your best friend when it comes to solving rational equations. It's the smallest expression that all denominators divide into evenly. Identifying the LCD is the foundation for clearing fractions and simplifying the equation. To find the LCD, factor each denominator completely. The LCD is then the product of the highest powers of all unique factors that appear in any of the denominators. For example, if your denominators are (x + 1) and (x + 1)(x - 2), the LCD is (x + 1)(x - 2). Remember, a clear understanding of how to find the LCD is crucial for tackling more complex rational equations with multiple terms.
2. Multiply Both Sides by the LCD
Once you've found the LCD, the next step is to multiply every term in the equation by it. This is the magic step that clears the fractions. When you multiply each term by the LCD, the denominators should cancel out, leaving you with a simpler equation to solve. Be meticulous in this step; make sure you distribute the LCD to each term correctly. A common mistake is forgetting to multiply a term by the LCD, which can lead to an incorrect solution. This step is vital because it transforms the rational equation into a more manageable form, often a linear or quadratic equation.
3. Simplify the Equation
After clearing the fractions, the next task is to simplify the equation. This usually involves distributing, combining like terms, and rearranging the equation into a standard form. If you end up with a linear equation, isolate the variable to solve. If you end up with a quadratic equation, you might need to factor, use the quadratic formula, or complete the square. The goal here is to reduce the equation to its simplest form so that you can easily identify the solution(s). Remember to double-check your work at each step to avoid errors in simplification. This step requires a solid understanding of algebraic manipulation, which is a fundamental skill in solving rational equations.
4. Solve the Resulting Equation
Once you've simplified the equation, you'll need to solve it for the variable. The method you use will depend on the type of equation you have. Linear equations can be solved by isolating the variable. Quadratic equations may require factoring, using the quadratic formula, or completing the square. For higher-degree polynomials, you might need to use more advanced techniques like synthetic division or the rational root theorem. Regardless of the method, make sure you show your work clearly and double-check your steps to avoid errors. This step is where your algebra skills truly shine, and a strong foundation in equation-solving techniques is essential for success with rational equations.
5. Check for Extraneous Solutions
This is arguably the most critical step in solving rational equations, and it's often overlooked. Extraneous solutions are solutions that you find algebraically, but they don't satisfy the original equation. These solutions typically arise when you've multiplied both sides of the equation by an expression that could be zero. To check for extraneous solutions, plug each solution you find back into the original equation. If a solution makes any of the denominators zero, it's an extraneous solution and must be discarded. Checking for extraneous solutions ensures that your answers are valid and that you haven't introduced any errors during the solving process. This step is a testament to the importance of rigor and precision in mathematics.
Real-World Applications of Rational Equations
Rational equations aren't just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can help you appreciate the practical significance of mastering these equations.
1. Physics
In physics, rational equations are used to describe relationships between physical quantities, such as speed, time, and distance. For instance, the equation speed = distance / time is a simple rational equation. More complex applications involve analyzing motion, forces, and energy. For example, in optics, the lens equation (1/f = 1/u + 1/v), which relates the focal length (f), object distance (u), and image distance (v) of a lens, is a rational equation. Solving these equations helps physicists make predictions and understand physical phenomena.
2. Engineering
Engineers use rational equations extensively in various disciplines. In electrical engineering, they're used to analyze circuits, calculate impedances, and determine current and voltage relationships. In civil engineering, they're used in structural analysis, fluid mechanics, and transportation planning. For example, the Manning equation, used to estimate the flow of water in open channels, is a rational equation. Chemical engineers use rational equations in reaction kinetics and process design. The ability to solve these equations is crucial for engineers to design efficient and safe systems.
3. Chemistry
Rational equations play a vital role in chemical kinetics and equilibrium. The rate of a chemical reaction can often be described by a rational equation, and the equilibrium constant for a reversible reaction is expressed as a ratio of concentrations. For example, the Michaelis-Menten equation, used in enzyme kinetics, is a rational equation that describes the rate of enzymatic reactions. Understanding and solving these equations is essential for chemists to study reaction mechanisms and predict reaction outcomes.
4. Economics
Economists use rational equations to model various economic phenomena. For example, supply and demand curves can be represented by rational equations, and the equilibrium price and quantity can be found by solving these equations. Rational equations are also used in cost-benefit analysis, financial modeling, and economic forecasting. The ability to work with these equations allows economists to analyze market behavior and make informed decisions.
5. Computer Science
In computer science, rational equations can be used in algorithm analysis and performance modeling. For example, the time complexity of an algorithm can sometimes be expressed as a rational equation. These equations help computer scientists understand how the performance of an algorithm scales with the input size. Rational equations are also used in networking to model network traffic and optimize network performance. The use of these equations allows for the design of efficient algorithms and systems.
Conclusion
So, guys, we've conquered another math challenge! We successfully solved the rational equation by eliminating fractions, simplifying, and checking for extraneous solutions. Remember, the key is to take it step by step, understand the underlying principles, and practice, practice, practice! Keep honing your skills, and you'll become a rational equation master in no time.
The solution set for the equation is {-9}.