What Is The Solution To The Equation 1 H − 5 + 2 H + 5 = 16 H 2 − 25 \frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25} H − 5 1 ​ + H + 5 2 ​ = H 2 − 25 16 ​ ?A. H = 11 3 H=\frac{11}{3} H = 3 11 ​ B. H = 5 H=5 H = 5 C. H = 7 H=7 H = 7 D. H = 21 2 H=\frac{21}{2} H = 2 21 ​

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Introduction

The equation $\frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25}$ is a rational equation that involves the sum of two fractions. To solve this equation, we need to find the value of $h$ that satisfies the equation. In this article, we will walk through the steps to solve the equation and find the solution.

Understanding the Equation

The given equation is $\frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25}$. The first step is to simplify the equation by finding a common denominator. The common denominator of the fractions on the left-hand side is $(h-5)(h+5)$.

Simplifying the Equation

To simplify the equation, we can multiply both sides of the equation by the common denominator $(h-5)(h+5)$. This will eliminate the fractions and make it easier to solve the equation.

\frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25}

Multiplying both sides by $(h-5)(h+5)$:

(h-5)(h+5)\left(\frac{1}{h-5}+\frac{2}{h+5}\right) = (h-5)(h+5)\left(\frac{16}{h^2-25}\right)

Simplifying the left-hand side:

(h+5) + 2(h-5) = 16

Expanding and simplifying:

h + 5 + 2h - 10 = 16

Combine like terms:

3h - 5 = 16

Solving for $h$

Now that we have simplified the equation, we can solve for $h$. To do this, we need to isolate $h$ on one side of the equation.

3h - 5 = 16

Adding 5 to both sides:

3h = 21

Dividing both sides by 3:

h = \frac{21}{3}

Simplifying:

h = 7

Checking the Solution

To check the solution, we can substitute $h=7$ back into the original equation and see if it is true.

\frac{1}{7-5}+\frac{2}{7+5}=\frac{16}{7^2-25}

Simplifying:

\frac{1}{2}+\frac{2}{12}=\frac{16}{49-25}

Simplifying further:

\frac{1}{2}+\frac{1}{6}=\frac{16}{24}

Combining fractions:

\frac{3}{6}+\frac{1}{6}=\frac{16}{24}

Simplifying:

\frac{4}{6}=\frac{16}{24}

Simplifying further:

\frac{2}{3}=\frac{16}{24}

Dividing both sides by 2:

\frac{1}{3}=\frac{8}{12}

Simplifying:

\frac{1}{3}=\frac{2}{3}

Since the equation is true, we have found the solution to the equation.

Conclusion

In this article, we solved the equation $\frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25}$ and found the solution to be $h=7$. We also checked the solution by substituting it back into the original equation and verified that it is true.

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Q: What is the first step in solving the equation $\frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25}$?

A: The first step in solving the equation is to simplify the equation by finding a common denominator. The common denominator of the fractions on the left-hand side is $(h-5)(h+5)$.

Q: How do I simplify the equation?

A: To simplify the equation, you can multiply both sides of the equation by the common denominator $(h-5)(h+5)$. This will eliminate the fractions and make it easier to solve the equation.

Q: What is the next step after simplifying the equation?

A: After simplifying the equation, you need to solve for $h$. To do this, you need to isolate $h$ on one side of the equation.

Q: How do I solve for $h$?

A: To solve for $h$, you need to isolate $h$ on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same non-zero value.

Q: What is the solution to the equation $\frac{1}{h-5}+\frac{2}{h+5}=\frac{16}{h^2-25}$?

A: The solution to the equation is $h=7$.

Q: How do I check the solution?

A: To check the solution, you can substitute $h=7$ back into the original equation and see if it is true.

Q: What if I get a different solution when I check it?

A: If you get a different solution when you check it, it means that the solution you found is not correct. You need to go back and recheck your work to find the correct solution.

Q: Can I use a calculator to solve the equation?

A: Yes, you can use a calculator to solve the equation. However, you need to make sure that you understand the steps involved in solving the equation and that you can explain the solution in your own words.

Q: What if I get stuck while solving the equation?

A: If you get stuck while solving the equation, you can try breaking down the problem into smaller steps or seeking help from a teacher or tutor.

Q: Can I use this method to solve other rational equations?

A: Yes, you can use this method to solve other rational equations. However, you need to make sure that you understand the steps involved in solving the equation and that you can explain the solution in your own words.

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

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

• Not simplifying the equation before solving it
• Not isolating the variable on one side of the equation
• Not checking the solution
• Not explaining the solution in your own words

Q: How can I practice solving rational equations?

A: You can practice solving rational equations by working on problems in your textbook or online resources. You can also try creating your own problems and solving them.

Q: What are some real-world applications of solving rational equations?

A: Solving rational equations has many real-world applications, including:

• Physics: Solving rational equations is used to describe the motion of objects and to calculate forces and energies.
• Engineering: Solving rational equations is used to design and optimize systems and to calculate stresses and strains.
• Economics: Solving rational equations is used to model economic systems and to calculate costs and benefits.

Q: Can I use technology to solve rational equations?

A: Yes, you can use technology to solve rational equations. However, you need to make sure that you understand the steps involved in solving the equation and that you can explain the solution in your own words.

Q: What are some benefits of using technology to solve rational equations?

A: Some benefits of using technology to solve rational equations include:

• Increased accuracy
• Faster solutions
• Ability to solve complex equations
• Ability to visualize the solution

Q: What are some limitations of using technology to solve rational equations?

A: Some limitations of using technology to solve rational equations include:

• Dependence on technology
• Limited understanding of the solution
• Limited ability to explain the solution in your own words

Q: Can I use this method to solve other types of equations?

A: Yes, you can use this method to solve other types of equations, including linear equations, quadratic equations, and polynomial equations. However, you need to make sure that you understand the steps involved in solving the equation and that you can explain the solution in your own words.