Solving Equations With Fractions A Step By Step Guide

$$

In the realm of mathematics, encountering equations involving fractions is a common occurrence. These equations, while seemingly complex at first glance, can be systematically solved by employing fundamental algebraic principles. This comprehensive guide delves into the step-by-step process of solving the equation ${\frac{6}{x-1}=\frac{9}{7}}$, providing a clear understanding of the underlying concepts and techniques.

1. Understanding the Problem: Deconstructing the Equation

Before embarking on the solution, it's crucial to grasp the essence of the equation. The equation ${\frac{6}{x-1}=\frac{9}{7}}$ presents a scenario where two fractions are equated. The variable 'x' resides in the denominator of the first fraction, making it an algebraic fraction. Our primary objective is to isolate 'x' and determine its value that satisfies the equation.

Key elements of the equation:

• Fractions: The equation comprises two fractions, each consisting of a numerator (the number above the line) and a denominator (the number below the line).
• Variable: The variable 'x' represents an unknown quantity that we aim to find.
• Equality: The equals sign signifies that the two fractions have the same value.

2. The Cross-Multiplication Technique: A Powerful Tool

The cross-multiplication technique serves as a cornerstone in solving equations involving fractions. This technique leverages the fundamental property that if two fractions are equal, then the product of the numerator of the first fraction and the denominator of the second fraction equals the product of the denominator of the first fraction and the numerator of the second fraction.

Applying cross-multiplication to our equation, we get:

${6 \times 7 = 9 \times (x-1)}$

This transformation eliminates the fractions, paving the way for a more straightforward algebraic manipulation.

3. Simplifying the Equation: Unveiling the Algebraic Structure

Now, let's simplify the equation obtained through cross-multiplication. We begin by performing the multiplications on both sides:

${42 = 9x - 9}$

Next, we aim to isolate the term containing 'x'. To achieve this, we add 9 to both sides of the equation:

${42 + 9 = 9x - 9 + 9}$

This simplifies to:

${51 = 9x}$

4. Isolating the Variable: The Final Step

To isolate 'x', we divide both sides of the equation by 9:

${\frac{51}{9} = \frac{9x}{9}}$

This yields:

${x = \frac{51}{9}}$

Simplifying the fraction, we get:

${x = \frac{17}{3}}$

Therefore, the solution to the equation ${\frac{6}{x-1}=\frac{9}{7}}$ is ${x = \frac{17}{3}}$.

5. Verification: Ensuring the Solution's Validity

To ensure the accuracy of our solution, we substitute ${x = \frac{17}{3}}$ back into the original equation:

${\frac{6}{\frac{17}{3}-1} = \frac{9}{7}}$

Simplifying the denominator of the first fraction:

${\frac{6}{\frac{17-3}{3}} = \frac{9}{7}}$

${\frac{6}{\frac{14}{3}} = \frac{9}{7}}$

Dividing by a fraction is equivalent to multiplying by its reciprocal:

${6 \times \frac{3}{14} = \frac{9}{7}}$

${\frac{18}{14} = \frac{9}{7}}$

Simplifying the fraction on the left-hand side:

${\frac{9}{7} = \frac{9}{7}}$

The equation holds true, confirming that our solution ${x = \frac{17}{3}}$ is indeed correct.

6. Alternative Approaches: Exploring Different Perspectives

While cross-multiplication provides an efficient method for solving equations involving fractions, alternative approaches can offer valuable insights and reinforce understanding. One such approach involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

In our equation, the denominators are (x-1) and 7. The LCM of these denominators is 7(x-1). Multiplying both sides of the equation by 7(x-1), we get:

${7(x-1) \times \frac{6}{x-1} = 7(x-1) \times \frac{9}{7}}$

Simplifying, we obtain:

${42 = 9(x-1)}$

This equation is identical to the one we arrived at through cross-multiplication, and the subsequent steps to solve for 'x' remain the same.

7. Potential Pitfalls: Avoiding Common Mistakes

When solving equations with fractions, several common pitfalls can lead to incorrect solutions. Awareness of these pitfalls is crucial for maintaining accuracy.

Pitfalls to watch out for:

• Dividing by Zero: Ensure that the denominator does not equal zero for any value of 'x'. In our equation, x cannot be equal to 1, as this would make the denominator of the first fraction zero.
• Incorrect Cross-Multiplication: Double-check the cross-multiplication step to ensure that the correct terms are multiplied.
• Arithmetic Errors: Pay close attention to arithmetic operations, especially when dealing with negative numbers or fractions.

8. Expanding the Knowledge: Beyond the Basics

The principles and techniques discussed in this guide extend to a wide range of equations involving fractions. As you delve deeper into mathematics, you'll encounter more complex equations, but the fundamental concepts remain the same. Mastering the art of solving equations with fractions lays a solid foundation for tackling advanced mathematical problems.

9. Real-World Applications: Connecting Math to Life

Equations involving fractions aren't confined to textbooks and classrooms. They pop up in various real-world scenarios, from calculating proportions in recipes to determining the speed of a vehicle. Understanding how to solve these equations equips you with valuable problem-solving skills that extend beyond the realm of mathematics.

10. Practice Makes Perfect: Hone Your Skills

The key to mastering any mathematical concept lies in consistent practice. Work through a variety of equations involving fractions, gradually increasing the complexity. The more you practice, the more confident and proficient you'll become in solving these equations.

In conclusion, solving the equation ${\frac{6}{x-1}=\frac{9}{7}}$ involves a systematic approach that combines algebraic principles with careful execution. By understanding the underlying concepts, mastering the cross-multiplication technique, and avoiding common pitfalls, you can confidently tackle a wide array of equations involving fractions. Remember, practice is the key to proficiency, so embrace the challenge and hone your skills to unlock the power of mathematics.

In the realm of mathematics, solving equations is a fundamental skill that underpins various concepts and applications. This guide focuses on a specific type of equation: those involving fractions. We will meticulously dissect the process of solving the equation ${\frac{6}{x-1}=\frac{9}{7}}$, providing a comprehensive understanding of each step involved. This equation falls under the category of algebraic fractions, where the variable 'x' appears in the denominator. Our primary goal is to isolate 'x' and determine its value that satisfies the given equality.

Understanding the Equation: A Foundation for Success

Before diving into the solution, it's essential to grasp the equation's structure and components. The equation ${\frac{6}{x-1}=\frac{9}{7}}$ presents a scenario where two fractions are set equal to each other. Each fraction consists of a numerator (the number above the line) and a denominator (the number below the line). The variable 'x' in the denominator of the first fraction introduces an algebraic element. The equals sign signifies that the two fractions represent the same numerical value.

Key Components of the Equation:

• Fractions: Two fractions are present, each representing a ratio of two numbers.
• Variable (x): The unknown quantity we aim to determine.
• Equality (=): The symbol indicating that both sides of the equation have the same value.
• Numerator: The top part of a fraction.
• Denominator: The bottom part of a fraction.

The Power of Cross-Multiplication: Simplifying the Equation

Cross-multiplication is a pivotal technique for solving equations involving fractions. This method stems from the principle that if two fractions are equal, then the product of the numerator of the first fraction and the denominator of the second fraction equals the product of the denominator of the first fraction and the numerator of the second fraction. In simpler terms, we multiply diagonally across the equals sign.

Applying Cross-Multiplication:

In our equation, we cross-multiply as follows:

${6 \times 7 = 9 \times (x-1)}$

This step effectively eliminates the fractions, transforming the equation into a more manageable algebraic expression.

Unveiling the Algebraic Structure: Simplification and Expansion

Now that we've eliminated the fractions, we can simplify the equation and prepare it for isolating 'x'. We begin by performing the multiplications on both sides:

${42 = 9x - 9}$

Next, we focus on isolating the term containing 'x'. To achieve this, we add 9 to both sides of the equation. This maintains the balance of the equation while moving the constant term to the left side:

${42 + 9 = 9x - 9 + 9}$

Simplifying, we get:

${51 = 9x}$

Isolating the Variable: The Final Calculation

The final step involves isolating 'x' to determine its value. To accomplish this, we divide both sides of the equation by 9:

${\frac{51}{9} = \frac{9x}{9}}$

This operation isolates 'x' on the right side, giving us:

${x = \frac{51}{9}}$

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3:

${x = \frac{17}{3}}$

Therefore, the solution to the equation ${\frac{6}{x-1}=\frac{9}{7}}$ is ${x = \frac{17}{3}}$. This value of 'x' satisfies the original equation, making both sides equal.

Verifying the Solution: Ensuring Accuracy

To ensure the correctness of our solution, we substitute ${x = \frac{17}{3}}$ back into the original equation and verify that both sides are indeed equal:

${\frac{6}{\frac{17}{3}-1} = \frac{9}{7}}$

Simplifying the denominator on the left side:

${\frac{6}{\frac{17-3}{3}} = \frac{9}{7}}$

${\frac{6}{\frac{14}{3}} = \frac{9}{7}}$

Dividing by a fraction is equivalent to multiplying by its reciprocal:

${6 \times \frac{3}{14} = \frac{9}{7}}$

${\frac{18}{14} = \frac{9}{7}}$

Simplifying the fraction on the left side:

${\frac{9}{7} = \frac{9}{7}}$

Since both sides of the equation are equal, our solution ${x = \frac{17}{3}}$ is verified.

Exploring Alternative Methods: A Broader Perspective

While cross-multiplication is an efficient technique, exploring alternative methods can enhance our understanding and problem-solving skills. One such approach involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

In our equation, the denominators are (x-1) and 7. The LCM of these denominators is 7(x-1). Multiplying both sides of the equation by 7(x-1), we get:

${7(x-1) \times \frac{6}{x-1} = 7(x-1) \times \frac{9}{7}}$

Simplifying, we obtain:

${42 = 9(x-1)}$

This equation is identical to the one we arrived at through cross-multiplication, and the subsequent steps to solve for 'x' remain the same. This alternative method reinforces the concept of manipulating equations while maintaining equality.

Common Pitfalls: Avoiding Errors in the Process

When solving equations involving fractions, several common pitfalls can lead to incorrect solutions. Awareness of these pitfalls is crucial for maintaining accuracy and avoiding mistakes.

Potential Pitfalls:

• Dividing by Zero: Ensure that the denominator does not equal zero for any value of 'x'. In our equation, x cannot be equal to 1, as this would make the denominator of the first fraction zero. Always check for values of 'x' that would result in division by zero and exclude them from the solution set.
• Incorrect Cross-Multiplication: Double-check the cross-multiplication step to ensure that the correct terms are multiplied. A simple mistake in this step can lead to an incorrect equation and an incorrect solution.
• Arithmetic Errors: Pay close attention to arithmetic operations, especially when dealing with negative numbers or fractions. Even a small arithmetic error can propagate through the solution and lead to an incorrect result. Use a calculator or double-check your calculations to minimize these errors.

Real-World Connections: Applications Beyond the Classroom

Equations involving fractions are not merely abstract mathematical concepts; they have practical applications in various real-world scenarios. Understanding how to solve these equations equips you with valuable problem-solving skills that extend beyond the classroom.

Real-World Applications:

• Proportions and Ratios: Many real-world problems involve proportions and ratios, which can be expressed as fractions. For example, scaling recipes, calculating exchange rates, or determining the concentration of a solution all involve equations with fractions.
• Physics and Engineering: Equations with fractions are prevalent in physics and engineering, particularly in areas such as mechanics, electricity, and fluid dynamics. For instance, calculating the resistance in a parallel circuit or determining the flow rate of a fluid through a pipe often involves solving equations with fractions.

By recognizing the real-world applications of equations with fractions, we can appreciate their significance and develop a deeper understanding of their utility.

In conclusion, solving the equation ${\frac{6}{x-1}=\frac{9}{7}}$ exemplifies the process of solving equations involving fractions. By understanding the structure of the equation, applying cross-multiplication, simplifying the expression, isolating the variable, verifying the solution, and exploring alternative methods, we can confidently tackle a wide range of similar problems. Moreover, being aware of potential pitfalls and recognizing the real-world connections of these equations enhances our problem-solving skills and mathematical literacy. Consistent practice and a thorough understanding of the underlying principles are key to mastering the art of solving equations with fractions.

In mathematics, the ability to solve equations is a fundamental skill, and equations involving fractions are a common occurrence. This article delves into the process of solving the specific equation ${\frac{6}{x-1}=\frac{9}{7}}$. This equation is an example of an algebraic fraction, where the unknown variable 'x' appears in the denominator. Our primary objective is to isolate 'x' and determine its value that makes the equation true. Understanding the steps involved in solving this equation provides a solid foundation for tackling more complex mathematical problems.

Deconstructing the Equation: A Step-by-Step Approach

Before we jump into the solution, let's break down the equation and understand its components. The equation ${\frac{6}{x-1}=\frac{9}{7}}$ presents a scenario where two fractions are equated. Each fraction has a numerator (the number on top) and a denominator (the number on the bottom). The variable 'x' resides in the denominator of the first fraction, which is what makes it an algebraic fraction. The equals sign indicates that the two fractions represent the same value.

Key components of the equation:

• Fractions: The equation involves two fractions, each representing a ratio between two numbers.
• Variable (x): The unknown value we are trying to find.
• Numerator: The number above the fraction line.
• Denominator: The number below the fraction line.
• Equality Sign (=): The symbol indicating that both sides of the equation have the same value.

Cross-Multiplication: A Powerful Technique for Solving Fractional Equations

The cornerstone of solving equations with fractions is a technique called cross-multiplication. This method is based on the principle that if two fractions are equal, then the product of the numerator of the first fraction and the denominator of the second fraction is equal to the product of the denominator of the first fraction and the numerator of the second fraction. Simply put, we multiply diagonally across the equals sign.

Applying cross-multiplication to our equation:

${6 \times 7 = 9 \times (x-1)}$

This step eliminates the fractions, making the equation easier to manipulate algebraically.

Simplifying the Equation: Unraveling the Algebraic Structure

With the fractions gone, we can now simplify the equation. First, we perform the multiplications:

${42 = 9x - 9}$

Our next goal is to isolate the term containing 'x'. To do this, we add 9 to both sides of the equation. This maintains the equality while moving the constant term to the left side:

${42 + 9 = 9x - 9 + 9}$

Simplifying, we get:

${51 = 9x}$

Isolating the Variable: Finding the Value of x

Now, we isolate 'x' by dividing both sides of the equation by 9:

${\frac{51}{9} = \frac{9x}{9}}$

This gives us:

${x = \frac{51}{9}}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

${x = \frac{17}{3}}$

Therefore, the solution to the equation ${\frac{6}{x-1}=\frac{9}{7}}$ is ${x = \frac{17}{3}}$.

Verifying the Solution: Ensuring Accuracy and Correctness

It's always a good practice to verify our solution by substituting it back into the original equation. Let's plug in ${x = \frac{17}{3}}$ into the equation ${\frac{6}{x-1}=\frac{9}{7}}$:

${\frac{6}{\frac{17}{3}-1} = \frac{9}{7}}$

First, we simplify the denominator on the left side:

${\frac{6}{\frac{17}{3}-\frac{3}{3}} = \frac{9}{7}}$

${\frac{6}{\frac{14}{3}} = \frac{9}{7}}$

Dividing by a fraction is the same as multiplying by its reciprocal:

${6 \times \frac{3}{14} = \frac{9}{7}}$

${\frac{18}{14} = \frac{9}{7}}$

Simplifying the fraction on the left:

${\frac{9}{7} = \frac{9}{7}}$

Since both sides are equal, our solution ${x = \frac{17}{3}}$ is correct.

Exploring Alternative Approaches: Expanding Our Mathematical Toolkit

While cross-multiplication is a highly efficient method for solving equations with fractions, exploring alternative approaches can enhance our understanding and problem-solving flexibility. One such method involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

In our equation, the denominators are (x-1) and 7. The LCM of these denominators is 7(x-1). Multiplying both sides of the equation by 7(x-1), we get:

${7(x-1) \times \frac{6}{x-1} = 7(x-1) \times \frac{9}{7}}$

Simplifying, we obtain:

${42 = 9(x-1)}$

This is the same equation we arrived at after cross-multiplication, and the subsequent steps to solve for 'x' are the same. This alternative approach reinforces the concept of manipulating equations while maintaining equality and highlights the versatility of mathematical techniques.

Potential Pitfalls: Avoiding Common Mistakes

When solving equations with fractions, there are several common pitfalls that can lead to incorrect solutions. Being aware of these potential errors can help us avoid them and ensure the accuracy of our solutions.

Common pitfalls to watch out for:

• Dividing by Zero: One of the most critical considerations in mathematics is the prohibition of division by zero. In our equation, we must ensure that the denominator (x-1) is not equal to zero. This means that x cannot be equal to 1. Always check for values that would make the denominator zero and exclude them from the solution set.
• Incorrect Cross-Multiplication: A simple mistake during cross-multiplication, such as multiplying the wrong terms, can lead to an incorrect equation and, consequently, an incorrect solution. It's crucial to double-check the cross-multiplication step to ensure that the correct terms are being multiplied.
• Arithmetic Errors: Errors in basic arithmetic operations, such as addition, subtraction, multiplication, or division, can also lead to incorrect solutions. Pay close attention to the signs of numbers and the order of operations. Using a calculator or double-checking calculations can help minimize these errors.

Real-World Applications: Connecting Mathematics to Our Lives

Equations involving fractions are not confined to the abstract world of mathematics; they have numerous practical applications in our daily lives. Understanding how to solve these equations equips us with valuable problem-solving skills that can be applied in various real-world scenarios.

Examples of real-world applications:

• Proportions and Ratios: Many real-world problems involve proportions and ratios, which can be expressed as fractions. For example, scaling a recipe, calculating currency exchange rates, determining the concentration of a solution, or figuring out the gear ratios in a bicycle all involve equations with fractions.
• Physics and Engineering: Equations with fractions are prevalent in physics and engineering, particularly in areas such as mechanics, electricity, and fluid dynamics. For instance, calculating the resistance in a parallel circuit, determining the flow rate of a fluid through a pipe, or analyzing the forces acting on an object often involves solving equations with fractions.

By recognizing the relevance of mathematics in our daily lives, we can develop a greater appreciation for its power and utility.

In conclusion, solving the equation ${\frac{6}{x-1}=\frac{9}{7}}$ demonstrates a fundamental process in algebra. By understanding the components of the equation, applying the technique of cross-multiplication, simplifying the resulting expression, isolating the variable, verifying the solution, and exploring alternative approaches, we can confidently solve a wide range of similar problems. Being mindful of potential pitfalls and recognizing the real-world applications of these equations further enhances our problem-solving skills and mathematical literacy. Consistent practice and a solid grasp of the underlying principles are essential for mastering the art of solving equations with fractions.