Solve The Equation, And Enter The 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-10}=\frac{x-9}{x-10} \ x=\square \text{ Or }

Mar 13, 2025 by ADMIN 224 views

Solve the Equation and Enter the Solutions from Least to Greatest

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Introduction

In this article, we will be solving a mathematical equation involving fractions and variables. The equation is given as $\frac{1}{x}+\frac{1}{x-10}=\frac{x-9}{x-10}$ . Our goal is to solve for the variable $x$ and enter the solutions from least to greatest. If there is only one solution, we will enter "n.a." for the second solution.

Step 1: Multiply Both Sides by the Common Denominator

To eliminate the fractions, we will multiply both sides of the equation by the common denominator, which is $(x)(x-10)$ . This will give us:

\frac{(x-10)}{x}+\frac{x}{x-10}=(x-9)

Step 2: Simplify the Equation

We can simplify the equation by multiplying both sides by $x(x-10)$ :

x(x-10)+x(x-10)=(x-9)x(x-10)

This simplifies to:

x^2-10x+x^2-10x=(x^2-9x-90)

Step 3: Combine Like Terms

We can combine like terms on the left-hand side of the equation:

2x^2-20x=(x^2-9x-90)

Step 4: Move All Terms to One Side

We will move all terms to the left-hand side of the equation by subtracting $(x^2-9x-90)$ from both sides:

2x^2-20x-(x^2-9x-90)=0

This simplifies to:

x^2-11x-90=0

Step 5: Factor the Quadratic Equation

We can factor the quadratic equation as:

(x-18)(x+5)=0

Step 6: Solve for x

We can solve for $x$ by setting each factor equal to zero:

x-18=0 \quad \text{or} \quad x+5=0

This gives us two possible solutions:

x=18 \quad \text{or} \quad x=-5

Conclusion

In conclusion, the solutions to the equation $\frac{1}{x}+\frac{1}{x-10}=\frac{x-9}{x-10}$ are $x=18$ and $x=-5$ . Since we are asked to enter the solutions from least to greatest, the final answer is:

-5, 18

Discussion

The equation we solved is a quadratic equation, which is a polynomial equation of degree two. Quadratic equations can be factored or solved using the quadratic formula. In this case, we were able to factor the equation and find the solutions.

Applications

The equation we solved has many real-world applications. For example, it can be used to model the motion of an object under the influence of gravity. The equation can also be used to model the growth of a population over time.

Future Work

In the future, we can use this equation to model more complex systems. For example, we can use it to model the motion of a pendulum or the growth of a population under the influence of multiple factors.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak

Keywords

Quadratic equation
Factoring
Solutions
Least to greatest
Algebra
Trigonometry
Calculus

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Introduction

In the previous article, we solved the equation $\frac{1}{x}+\frac{1}{x-10}=\frac{x-9}{x-10}$ and found the solutions to be $x=18$ and $x=-5$ . In this article, we will answer some frequently asked questions (FAQs) about solving the equation.

Q: What is the purpose of multiplying both sides of the equation by the common denominator?

A: The purpose of multiplying both sides of the equation by the common denominator is to eliminate the fractions. This makes it easier to simplify the equation and solve for the variable.

Q: How do I know which terms to combine when simplifying the equation?

A: When simplifying the equation, you should combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in the equation $x^2-10x+x^2-10x=(x^2-9x-90)$ , the like terms are $x^2$ and $-10x$ .

Q: What is the difference between factoring and solving a quadratic equation?

A: Factoring and solving a quadratic equation are two different methods of finding the solutions to a quadratic equation. Factoring involves expressing the quadratic equation as a product of two binomials, while solving a quadratic equation involves using the quadratic formula or other methods to find the solutions.

Q: Can I use the quadratic formula to solve the equation?

A: Yes, you can use the quadratic formula to solve the equation. The quadratic formula is given by:

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

In this case, the equation is $x^2-11x-90=0$ , so we can plug in the values of $a$ , $b$ , and $c$ into the quadratic formula to find the solutions.

Q: What is the significance of the solutions to the equation?

A: The solutions to the equation represent the values of the variable that satisfy the equation. In this case, the solutions are $x=18$ and $x=-5$ . These values represent the points at which the equation is true.

Q: Can I use the equation to model real-world situations?

A: Yes, you can use the equation to model real-world situations. For example, the equation can be used to model the motion of an object under the influence of gravity or the growth of a population over time.

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

A: Some common mistakes to avoid when solving the equation include:

Not multiplying both sides of the equation by the common denominator
Not combining like terms when simplifying the equation
Not factoring the quadratic equation correctly
Not using the quadratic formula correctly

Conclusion

In conclusion, solving the equation $\frac{1}{x}+\frac{1}{x-10}=\frac{x-9}{x-10}$ requires careful attention to detail and a thorough understanding of algebraic concepts. By following the steps outlined in this article, you can solve the equation and find the solutions.