Solve The Equation: X + 3 X − 1 = X + 3 3 \frac{x+3}{x-1}=\frac{x+3}{3} X − 1 X + 3 ​ = 3 X + 3 ​ A. X = 1 ; X = − 3 X=1 ; X=-3 X = 1 ; X = − 3 Is ExtraneousB. X = 1 ; X = − 3 X=1 ; X=-3 X = 1 ; X = − 3 C. X = 4 ; X = − 3 X=4 ; X=-3 X = 4 ; X = − 3 D. X = 4 ; X = − 3 X=4 ; X=-3 X = 4 ; X = − 3 Is Extraneous

Introduction

In this article, we will be solving a complex equation involving fractions. The equation is given as $\frac{x+3}{x-1}=\frac{x+3}{3}$. Our goal is to find the value of $x$ that satisfies this equation. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Multiply Both Sides by the Least Common Denominator (LCD)

The first step in solving this equation is to eliminate the fractions. To do this, we need to multiply both sides of the equation by the least common denominator (LCD), which is $(x-1)(3)$. This will allow us to work with whole numbers and simplify the equation.

$\frac{x+3}{x-1}=\frac{x+3}{3}$

Multiplying both sides by $(x-1)(3)$:

$(x-1)(3)\cdot\frac{x+3}{x-1}=(x-1)(3)\cdot\frac{x+3}{3}$

Simplifying the equation:

$3(x+3)=3(x+3)$

Step 2: Simplify the Equation

Now that we have eliminated the fractions, we can simplify the equation by canceling out the common terms.

$3(x+3)=3(x+3)$

Subtracting $3(x+3)$ from both sides:

$0=0$

This equation is true for all values of $x$, which means that the original equation is an identity. However, we need to be careful and check for any extraneous solutions.

Step 3: Check for Extraneous Solutions

An extraneous solution is a value of $x$ that satisfies the original equation but does not satisfy the simplified equation. In this case, we need to check if $x=1$ and $x=-3$ are extraneous solutions.

Substituting $x=1$ into the original equation:

$\frac{1+3}{1-1}=\frac{1+3}{3}$

This equation is undefined, which means that $x=1$ is an extraneous solution.

Substituting $x=-3$ into the original equation:

$\frac{-3+3}{-3-1}=\frac{-3+3}{3}$

This equation is true, which means that $x=-3$ is not an extraneous solution.

Conclusion

In conclusion, the solution to the equation $\frac{x+3}{x-1}=\frac{x+3}{3}$ is $x=-3$. The value $x=1$ is an extraneous solution, which means that it does not satisfy the original equation.

Final Answer

The final answer is:

• $x=-3$
• $x=1$ is extraneous

Discussion

This problem requires a good understanding of algebraic manipulations and the concept of extraneous solutions. It is essential to be careful when simplifying equations and to check for any extraneous solutions.

Related Problems

• Solve the equation $\frac{x-2}{x+1}=\frac{x-2}{2}$
• Find the value of $x$ that satisfies the equation $\frac{x+1}{x-2}=\frac{x+1}{3}$

Additional Resources

• Algebraic manipulations
• Extraneous solutions
• Equations with fractions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Solving the Equation: A Q&A Guide =====================================

Introduction

In our previous article, we solved the equation $\frac{x+3}{x-1}=\frac{x+3}{3}$ and found that the solution is $x=-3$. However, we also found that $x=1$ is an extraneous solution. In this article, we will answer some frequently asked questions about solving equations with fractions and extraneous solutions.

Q: What is an extraneous solution?

A: An extraneous solution is a value of $x$ that satisfies the original equation but does not satisfy the simplified equation. In other words, it is a solution that is not valid.

Q: How do I know if a solution is extraneous?

A: To determine if a solution is extraneous, you need to substitute the value of $x$ into the original equation and check if it is true. If the equation is undefined or false, then the solution is extraneous.

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

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

• Not multiplying both sides of the equation by the least common denominator (LCD)
• Not simplifying the equation correctly
• Not checking for extraneous solutions

Q: How do I multiply both sides of an equation by the LCD?

A: To multiply both sides of an equation by the LCD, you need to multiply each term on both sides of the equation by the LCD. For example, if the equation is $\frac{x+3}{x-1}=\frac{x+3}{3}$, the LCD is $(x-1)(3)$, so you would multiply both sides by $(x-1)(3)$.

Q: What is the least common denominator (LCD)?

A: The least common denominator (LCD) is the smallest multiple of all the denominators in an equation. For example, if the equation is $\frac{x+3}{x-1}=\frac{x+3}{3}$, the LCD is $(x-1)(3)$.

Q: How do I simplify an equation with fractions?

A: To simplify an equation with fractions, you need to cancel out any common terms on both sides of the equation. For example, if the equation is $\frac{x+3}{x-1}=\frac{x+3}{3}$, you can cancel out the common term $(x+3)$ on both sides.

Q: What are some common types of equations with fractions?

A: Some common types of equations with fractions include:

• Equations with linear fractions (e.g. $\frac{x+3}{x-1}=\frac{x+3}{3}$)
• Equations with quadratic fractions (e.g. $\frac{x^2+3x}{x^2-1}=\frac{x^2+3x}{x+1}$)
• Equations with rational expressions (e.g. $\frac{x+3}{x-1}=\frac{x+3}{3x}$)

Conclusion

In conclusion, solving equations with fractions requires careful attention to detail and a good understanding of algebraic manipulations. By following the steps outlined in this article, you can solve equations with fractions and avoid common mistakes.

Final Tips

• Always multiply both sides of the equation by the LCD
• Simplify the equation correctly
• Check for extraneous solutions
• Practice solving equations with fractions to build your skills and confidence.

Related Resources

• Algebraic manipulations
• Extraneous solutions
• Equations with fractions
• Least common denominator (LCD)
• Simplifying equations with fractions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton