Type The Correct Answer In Each Box.Solve The Equation, And Enter Solutions From Least To Greatest. If There Is Only One Solution, Enter n.a. For The Second Solution.$\[ \begin{array}{l} \frac{1}{x}+\frac{1}{x-3}=\frac{x-2}{x-3} \\ x=\quad

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Introduction

In this article, we will be solving a linear equation involving fractions. The equation is given as 1x+1x−3=x−2x−3\frac{1}{x}+\frac{1}{x-3}=\frac{x-2}{x-3}. Our goal is to find the solutions to this equation and present them in order from least to greatest.

Step 1: Multiply Both Sides by the Common Denominator

To eliminate the fractions, we need to multiply both sides of the equation by the common denominator, which is (x)(x−3)(x)(x-3). This will allow us to work with polynomials instead of fractions.

\frac{1}{x}+\frac{1}{x-3}=\frac{x-2}{x-3}
(x)(x-3)\left(\frac{1}{x}+\frac{1}{x-3}\right) = (x)(x-3)\left(\frac{x-2}{x-3}\right)

Step 2: Simplify the Equation

After multiplying both sides by the common denominator, we can simplify the equation by canceling out the common factors.

(x-3) + x = x(x-2)
x^2 - 2x + x - 3 = x^2 - 2x
x^2 - x - 3 = 0

Step 3: Solve the Quadratic Equation

Now that we have a quadratic equation, we can use the quadratic formula to find the solutions.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-3)}}{2(1)}
x = \frac{1 \pm \sqrt{1 + 12}}{2}
x = \frac{1 \pm \sqrt{13}}{2}

Step 4: Present the Solutions in Order from Least to Greatest

Since we have two solutions, we need to present them in order from least to greatest. We can do this by comparing the values of the two solutions.

x_1 = \frac{1 - \sqrt{13}}{2}
x_2 = \frac{1 + \sqrt{13}}{2}

Since 13>1\sqrt{13} > 1, we know that x2>x1x_2 > x_1. Therefore, the solutions to the equation are:

x_1 = \frac{1 - \sqrt{13}}{2}
x_2 = \frac{1 + \sqrt{13}}{2}

Conclusion

In this article, we solved a linear equation involving fractions and presented the solutions in order from least to greatest. We used the quadratic formula to find the solutions and compared the values to determine the order.

Final Answer

The final answer is:

Box Solution
1 1−132\frac{1 - \sqrt{13}}{2}
2 1+132\frac{1 + \sqrt{13}}{2}

Note: If there is only one solution, enter "n.a." for the second solution.

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Introduction

In our previous article, we solved a linear equation involving fractions and presented the solutions in order from least to greatest. In this article, we will answer some common questions related to solving the equation.

Q: What is the common denominator in the equation?

A: The common denominator in the equation is (x)(x−3)(x)(x-3).

Q: Why do we need to multiply both sides by the common denominator?

A: We need to multiply both sides by the common denominator to eliminate the fractions and work with polynomials instead of fractions.

Q: How do we simplify the equation after multiplying both sides by the common denominator?

A: After multiplying both sides by the common denominator, we can simplify the equation by canceling out the common factors.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to find the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do we use the quadratic formula to find the solutions to the equation?

A: To use the quadratic formula, we need to identify the values of aa, bb, and cc in the quadratic equation. In this case, a=1a = 1, b=−1b = -1, and c=−3c = -3. We can then plug these values into the quadratic formula to find the solutions.

Q: What are the solutions to the equation?

A: The solutions to the equation are:

x1=1−132x_1 = \frac{1 - \sqrt{13}}{2}

x2=1+132x_2 = \frac{1 + \sqrt{13}}{2}

Q: How do we present the solutions in order from least to greatest?

A: To present the solutions in order from least to greatest, we need to compare the values of the two solutions. Since 13>1\sqrt{13} > 1, we know that x2>x1x_2 > x_1. Therefore, the solutions to the equation are:

x1=1−132x_1 = \frac{1 - \sqrt{13}}{2}

x2=1+132x_2 = \frac{1 + \sqrt{13}}{2}

Q: What is the final answer to the equation?

A: The final answer to the equation is:

Box Solution
1 1−132\frac{1 - \sqrt{13}}{2}
2 1+132\frac{1 + \sqrt{13}}{2}

Note: If there is only one solution, enter "n.a." for the second solution.

Conclusion

In this article, we answered some common questions related to solving the equation. We hope that this Q&A guide has been helpful in understanding the solution to the equation.

Related Articles

  • Solving the Equation: A Step-by-Step Guide
  • Solving the Equation: A Guide to the Quadratic Formula

Tags

  • Solving the Equation
  • Quadratic Formula
  • Linear Equation
  • Fractions
  • Polynomials
  • Quadratic Equation

Categories

  • Mathematics
  • Algebra
  • Equations
  • Quadratic Equations
  • Linear Equations