Solve The Equation:$\[ \frac{3}{5} + \frac{1}{x} = \frac{2}{3} \\]

Introduction

In this article, we will delve into the world of mathematics and explore a simple yet intriguing equation. The equation in question is $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$, and our goal is to solve for the variable $x$. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and find the solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what we're dealing with. The equation is a linear equation, which means it can be represented in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants. In this case, the equation is $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To solve for $x$, we need to get rid of the fractions in the equation. One way to do this is to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of $5$, $x$, and $3$ is $15x$. Multiplying both sides of the equation by $15x$ gives us:

$$15x \left(\frac{3}{5} + \frac{1}{x}\right) = 15x \left(\frac{2}{3}\right)$$

Step 2: Distribute the Multiplication

Now that we have multiplied both sides of the equation by $15x$, we can distribute the multiplication to simplify the equation. This gives us:

$$9x + 15 = 10x$$

Step 3: Isolate the Variable

Our goal is to isolate the variable $x$ on one side of the equation. To do this, we can subtract $9x$ from both sides of the equation, which gives us:

$$15 = x$$

Conclusion

And there you have it! We have successfully solved the equation $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$ for the variable $x$. The solution is $x = 15$. This equation may have seemed daunting at first, but by breaking it down into smaller steps and using algebraic techniques, we were able to find the solution.

Real-World Applications

Solving equations like this one has many real-world applications. For example, in physics, we often encounter equations that involve fractions and variables. By learning how to solve equations like this one, we can apply these skills to solve problems in physics and other fields.

Tips and Tricks

Here are a few tips and tricks to keep in mind when solving equations like this one:

• Use the least common multiple (LCM): When multiplying both sides of the equation by the LCM, make sure to multiply both sides by the same value.
• Distribute the multiplication: When distributing the multiplication, make sure to multiply each term on the left-hand side of the equation by the same value.
• Isolate the variable: When isolating the variable, make sure to subtract or add the same value to both sides of the equation.

Common Mistakes

Here are a few common mistakes to avoid when solving equations like this one:

• Not using the LCM: Failing to use the LCM can lead to incorrect solutions.
• Not distributing the multiplication: Failing to distribute the multiplication can lead to incorrect solutions.
• Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.

Conclusion

Introduction

In our previous article, we explored the equation $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$ and provided a step-by-step guide on how to solve for the variable $x$. In this article, we will continue to delve into the world of mathematics and provide a Q&A guide on solving equations like this one.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the case of the equation $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$, the LCM is $15x$.

Q: Why do we need to multiply both sides of the equation by the LCM?

A: We need to multiply both sides of the equation by the LCM to get rid of the fractions in the equation. This allows us to simplify the equation and solve for the variable $x$.

Q: What is the difference between the LCM and the greatest common divisor (GCD)?

A: The LCM and the GCD are two different concepts in mathematics. The GCD is the largest number that two or more numbers have in common, while the LCM is the smallest multiple that two or more numbers have in common.

Q: How do we distribute the multiplication in an equation?

A: When distributing the multiplication in an equation, we multiply each term on the left-hand side of the equation by the same value. For example, in the equation $2x + 3 = 5$, we would multiply each term by $2$ to get $4x + 6 = 10$.

Q: What is the importance of isolating the variable in an equation?

A: Isolating the variable in an equation is crucial because it allows us to solve for the variable and find its value. In the equation $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$, isolating the variable $x$ allows us to find its value and solve the equation.

Q: What are some common mistakes to avoid when solving equations?

A: Some common mistakes to avoid when solving equations include:

• Not using the LCM
• Not distributing the multiplication
• Not isolating the variable
• Not checking the solution

Q: How do we check the solution to an equation?

A: To check the solution to an equation, we substitute the value of the variable back into the original equation and see if it is true. For example, in the equation $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$, we would substitute $x = 15$ back into the equation and see if it is true.

Q: What are some real-world applications of solving equations?

A: Solving equations has many real-world applications, including:

• Physics: Solving equations is crucial in physics to describe the motion of objects and predict their behavior.
• Engineering: Solving equations is used in engineering to design and optimize systems.
• Economics: Solving equations is used in economics to model economic systems and make predictions about economic behavior.

Conclusion

Solving equations like $\frac{3}{5} + \frac{1}{x} = \frac{2}{3}$ requires a clear understanding of algebraic techniques and a step-by-step approach. By following the steps outlined in this article and avoiding common mistakes, we can solve equations like this one and apply these skills to solve problems in physics, engineering, and economics.