Solve The Rational Equation. Express Numbers As Integers Or Simplified Fractions.${ \frac{1}{5} + \frac{4}{w-3} = 1 }$The Solution Set Is { {\ \square\ }$}$.

by ADMIN 159 views

=====================================================

Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a clear understanding of fractions, variables, and algebraic manipulations. In this article, we will focus on solving a specific rational equation, 15+4w−3=1\frac{1}{5} + \frac{4}{w-3} = 1, and express the solution set in integers or simplified fractions.

Understanding Rational Equations

A rational equation is an equation that contains one or more rational expressions, which are fractions with polynomials in the numerator and denominator. Rational equations can be solved using various techniques, including finding a common denominator, multiplying both sides by the common denominator, and simplifying the resulting expression.

Step 1: Multiply Both Sides by the Common Denominator

To solve the given rational equation, we need to eliminate the fractions by multiplying both sides by the common denominator, which is (w−3)(w-3). This will allow us to simplify the equation and solve for the variable ww.

\frac{1}{5} + \frac{4}{w-3} = 1
\\
\Rightarrow \qquad \frac{1}{5}(w-3) + 4 = (w-3)
\\
\Rightarrow \qquad \frac{w-3}{5} + 4 = w-3

Step 2: Simplify the Equation

Now that we have multiplied both sides by the common denominator, we can simplify the equation by combining like terms.

\frac{w-3}{5} + 4 = w-3
\\
\Rightarrow \qquad \frac{w-3}{5} - w + 3 = 0
\\
\Rightarrow \qquad \frac{w-3}{5} - \frac{5w-15}{5} = 0
\\
\Rightarrow \qquad \frac{w-3-5w+15}{5} = 0
\\
\Rightarrow \qquad \frac{-4w+12}{5} = 0

Step 3: Solve for the Variable

To solve for the variable ww, we need to isolate it on one side of the equation. In this case, we can multiply both sides by 55 to eliminate the fraction.

\frac{-4w+12}{5} = 0
\\
\Rightarrow \qquad -4w+12 = 0
\\
\Rightarrow \qquad -4w = -12
\\
\Rightarrow \qquad w = 3

Conclusion

In this article, we have solved a rational equation, 15+4w−3=1\frac{1}{5} + \frac{4}{w-3} = 1, and expressed the solution set in integers or simplified fractions. We have used various techniques, including finding a common denominator, multiplying both sides by the common denominator, and simplifying the resulting expression. The solution to the equation is w=3w = 3.

Final Answer

The final answer is 3\boxed{3}.

Discussion

Rational equations are a fundamental concept in algebra, and solving them requires a clear understanding of fractions, variables, and algebraic manipulations. In this article, we have focused on solving a specific rational equation, 15+4w−3=1\frac{1}{5} + \frac{4}{w-3} = 1, and expressed the solution set in integers or simplified fractions. We have used various techniques, including finding a common denominator, multiplying both sides by the common denominator, and simplifying the resulting expression. The solution to the equation is w=3w = 3.

Related Topics

  • Solving Linear Equations
  • Solving Quadratic Equations
  • Solving Polynomial Equations
  • Rational Expressions
  • Algebraic Manipulations

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for general information purposes only and are not directly related to the specific rational equation solved in this article.

=====================================

Introduction

In our previous article, we solved a rational equation, 15+4w−3=1\frac{1}{5} + \frac{4}{w-3} = 1, and expressed the solution set in integers or simplified fractions. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving rational equations.

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are fractions with polynomials in the numerator and denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to follow these steps:

  1. Find a common denominator for all the rational expressions in the equation.
  2. Multiply both sides of the equation by the common denominator to eliminate the fractions.
  3. Simplify the resulting expression and solve for the variable.

Q: What is a common denominator?

A: A common denominator is a polynomial that is a multiple of all the denominators in the rational expressions in the equation.

Q: How do I find a common denominator?

A: To find a common denominator, you need to identify the denominators in the rational expressions and find the least common multiple (LCM) of all the denominators.

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

A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of all the numbers in the set.

Q: How do I multiply both sides of the equation by the common denominator?

A: To multiply both sides of the equation by the common denominator, you need to multiply each term in the equation by the common denominator.

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

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

  • Not finding a common denominator
  • Not multiplying both sides of the equation by the common denominator
  • Not simplifying the resulting expression
  • Not solving for the variable

Q: What are some tips for solving rational equations?

A: Some tips for solving rational equations include:

  • Start by finding a common denominator
  • Multiply both sides of the equation by the common denominator
  • Simplify the resulting expression
  • Solve for the variable
  • Check your solution by plugging it back into the original equation

Q: How do I check my solution?

A: To check your solution, you need to plug it back into the original equation and verify that it is true.

Q: What are some common applications of rational equations?

A: Some common applications of rational equations include:

  • Solving systems of equations
  • Finding the intersection of two curves
  • Solving optimization problems
  • Finding the area of a region

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concepts and techniques involved in solving rational equations. We have covered topics such as finding a common denominator, multiplying both sides of the equation by the common denominator, and simplifying the resulting expression. We have also provided some tips and common mistakes to avoid when solving rational equations.

Final Answer

The final answer is 3\boxed{3}.

Discussion

Rational equations are a fundamental concept in algebra, and solving them requires a clear understanding of fractions, variables, and algebraic manipulations. In this article, we have focused on providing a Q&A guide to help you better understand the concepts and techniques involved in solving rational equations.

Related Topics

  • Solving Linear Equations
  • Solving Quadratic Equations
  • Solving Polynomial Equations
  • Rational Expressions
  • Algebraic Manipulations

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for general information purposes only and are not directly related to the specific rational equation solved in this article.