Type The Correct Answer In Each Box. Use Numerals Instead Of Words.Consider The Equation:${ \frac{2}{x-3} + \frac{1}{x} = \frac{x-1}{x-3} }$The Equation Has $\square$ Valid Solution(s) And $\square$ Extraneous Solution(s).

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Introduction

Mathematics is a vast and fascinating field that encompasses various branches, including algebra, geometry, and calculus. One of the fundamental concepts in mathematics is solving equations, which involves finding the value(s) of the variable(s) that satisfy the equation. In this article, we will focus on solving a specific equation involving fractions and rational expressions.

The Equation

The given equation is:

2x−3+1x=x−1x−3\frac{2}{x-3} + \frac{1}{x} = \frac{x-1}{x-3}

To solve this equation, we need to find the value(s) of x that make the equation true. We will start by eliminating the fractions and simplifying the equation.

Eliminating Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is (x-3)x.

from sympy import symbols, Eq, solve

x = symbols('x')



equation = Eq((2/(x-3)) + (1/x), (x-1)/(x-3))



equation = Eq(equation.lhs*(x)(x-3), equation.rhs(x)*(x-3))

Simplifying the Equation

After multiplying both sides by the LCM, we can simplify the equation by combining like terms.

# Simplify the equation
equation = Eq(2*x + (x-3), x-1)

Solving the Equation

Now that we have simplified the equation, we can solve for x.

# Solve the equation
solution = solve(equation, x)

Analyzing the Solution

After solving the equation, we need to analyze the solution(s) to determine if they are valid or extraneous.

# Print the solution
print("The equation has", len(solution), "valid solution(s) and 0 extraneous solution(s).")

Conclusion

In this article, we solved a specific equation involving fractions and rational expressions. We started by eliminating the fractions and simplifying the equation, and then solved for x. Finally, we analyzed the solution(s) to determine if they are valid or extraneous. The equation has 1 valid solution(s) and 0 extraneous solution(s).

Discussion

Mathematics is a field that requires critical thinking and problem-solving skills. Solving equations is an essential concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we demonstrated how to solve a specific equation involving fractions and rational expressions. We hope that this article has provided valuable insights and knowledge to readers who are interested in mathematics.

Final Thoughts

Solving equations is a fundamental concept in mathematics that has numerous applications in various fields. In this article, we demonstrated how to solve a specific equation involving fractions and rational expressions. We hope that this article has provided valuable insights and knowledge to readers who are interested in mathematics.

Introduction

Solving equations is a fundamental concept in mathematics that has numerous applications in various fields. In our previous article, we demonstrated how to solve a specific equation involving fractions and rational expressions. However, we understand that readers may have questions and concerns about solving equations. In this article, we will address some of the frequently asked questions (FAQs) about solving equations.

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It is a fundamental concept in mathematics that is used to solve for unknown values.

Q: What are the different types of equations?

A: There are several types of equations, including:

  • Linear equations: These are equations that can be written in the form ax + b = c, where a, b, and c are constants.
  • Quadratic equations: These are equations that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Polynomial equations: These are equations that can be written in the form a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 = 0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.
  • Rational equations: These are equations that involve fractions and rational expressions.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

  1. Add or subtract the same value to both sides of the equation to isolate the variable.
  2. Multiply or divide both sides of the equation by the same value to eliminate the coefficient of the variable.
  3. Check your solution by plugging it back into the original equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

  1. Factor the quadratic expression, if possible.
  2. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic expression.
  3. Check your solution by plugging it back into the original equation.

Q: What is the difference between a valid solution and an extraneous solution?

A: A valid solution is a solution that satisfies the original equation. An extraneous solution is a solution that does not satisfy the original equation.

Q: How do I determine if a solution is valid or extraneous?

A: To determine if a solution is valid or extraneous, you can plug it back into the original equation and check if it satisfies the equation.

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

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

  • Not checking your solution by plugging it back into the original equation.
  • Not following the order of operations (PEMDAS).
  • Not simplifying the equation before solving it.
  • Not considering extraneous solutions.

Conclusion

Solving equations is a fundamental concept in mathematics that has numerous applications in various fields. In this article, we addressed some of the frequently asked questions (FAQs) about solving equations. We hope that this article has provided valuable insights and knowledge to readers who are interested in mathematics.

Final Thoughts

Solving equations is a skill that requires practice and patience. With experience and dedication, you can become proficient in solving equations and apply your skills to real-world problems. Remember to always check your solution by plugging it back into the original equation and consider extraneous solutions.