Solve The Rational Equation. If There Is No Solution, Enter DNE. − 5 P − 3 + 10 P − 7 = 1 -\frac{5}{p-3}+\frac{10}{p-7}=1 − P − 3 5 ​ + P − 7 10 ​ = 1 Solution: P = P= P = □ \square □ If There Is More Than One Solution To The Equation, Use A Comma To Separate Solutions.

Introduction

Rational equations are a type of algebraic equation that involves fractions with polynomials in the numerator and denominator. Solving rational equations can be challenging, but with the right approach, it can be done efficiently. In this article, we will focus on solving the rational equation $-\frac{5}{p-3}+\frac{10}{p-7}=1$. We will break down the solution step by step and provide a clear explanation of each step.

Step 1: Write Down the Given Equation

The given equation is $-\frac{5}{p-3}+\frac{10}{p-7}=1$. Our goal is to solve for $p$.

Step 2: Eliminate the Fractions

To eliminate the fractions, we need to find the least common multiple (LCM) of the denominators, which are $p-3$ and $p-7$. The LCM of these two expressions is $(p-3)(p-7)$.

Step 3: Multiply Both Sides of the Equation by the LCM

We multiply both sides of the equation by $(p-3)(p-7)$ to eliminate the fractions.

$$-\frac{5}{p-3}\cdot(p-3)(p-7)+\frac{10}{p-7}\cdot(p-3)(p-7)=(p-3)(p-7)$$

This simplifies to:

$$-5(p-7)+10(p-3)=(p-3)(p-7)$$

Step 4: Expand and Simplify the Equation

We expand and simplify the equation by distributing the negative sign and combining like terms.

$$-5p+35+10p-30=p^2-10p+21$$

This simplifies to:

$$5p+5=p^2-10p+21$$

Step 5: Rearrange the Equation

We rearrange the equation to get a quadratic equation in standard form.

$$p^2-15p+16=0$$

Step 6: Factor the Quadratic Equation

We factor the quadratic equation to find the solutions for $p$.

$$(p-16)(p-1)=0$$

Step 7: Solve for $p$

We solve for $p$ by setting each factor equal to zero and solving for $p$.

$$p-16=0 \Rightarrow p=16$$

$$p-1=0 \Rightarrow p=1$$

Conclusion

The solutions to the rational equation $-\frac{5}{p-3}+\frac{10}{p-7}=1$ are $p=16$ and $p=1$. These solutions satisfy the original equation.

Discussion

Solving rational equations can be challenging, but with the right approach, it can be done efficiently. In this article, we broke down the solution step by step and provided a clear explanation of each step. We also discussed the importance of eliminating fractions and rearranging the equation to get a quadratic equation in standard form. By following these steps, you can solve rational equations with confidence.

Tips and Tricks

• When solving rational equations, it's essential to eliminate fractions by finding the least common multiple (LCM) of the denominators.
• Rearranging the equation to get a quadratic equation in standard form can make it easier to solve.
• Factoring the quadratic equation can help you find the solutions for $p$.
• Be careful when solving for $p$ by setting each factor equal to zero and solving for $p$.

Common Mistakes

• Failing to eliminate fractions can lead to incorrect solutions.
• Not rearranging the equation to get a quadratic equation in standard form can make it difficult to solve.
• Not factoring the quadratic equation can make it challenging to find the solutions for $p$.

Real-World Applications

Rational equations have many real-world applications, including:

• Physics: Rational equations are used to describe the motion of objects under the influence of forces.
• Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Rational equations are used to model economic systems and make predictions about future trends.

Conclusion

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Introduction

Solving rational equations can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving rational equations.

Q: What is a rational equation?

A: A rational equation is an equation that involves fractions with polynomials in the numerator and denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to follow these steps:

1. Eliminate the fractions by finding the least common multiple (LCM) of the denominators.
2. Rearrange the equation to get a quadratic equation in standard form.
3. Factor the quadratic equation to find the solutions for the variable.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two or more expressions is the smallest expression that is a multiple of each of the given expressions.

Q: How do I find the LCM of two expressions?

A: To find the LCM of two expressions, you can list the multiples of each expression and find the smallest multiple that is common to both.

Q: What is a quadratic equation?

A: A quadratic equation is an equation of the form $ax^2+bx+c=0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two binomials whose product is equal to the quadratic expression.

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

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

• Failing to eliminate fractions
• Not rearranging the equation to get a quadratic equation in standard form
• Not factoring the quadratic equation

Q: How do I apply rational equations to real-world problems?

A: Rational equations have many real-world applications, including:

• Physics: Rational equations are used to describe the motion of objects under the influence of forces.
• Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Rational equations are used to model economic systems and make predictions about future trends.

Q: What are some tips and tricks for solving rational equations?

A: Some tips and tricks for solving rational equations include:

• Eliminating fractions by finding the LCM of the denominators
• Rearranging the equation to get a quadratic equation in standard form
• Factoring the quadratic equation to find the solutions for the variable

Q: How do I check my solutions to a rational equation?

A: To check your solutions to a rational equation, you need to substitute each solution back into the original equation and verify that it is true.

Conclusion

Solving rational equations can be a challenging task, but with the right approach, it can be done efficiently. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving rational equations and apply them to real-world problems. Remember to eliminate fractions, rearrange the equation to get a quadratic equation in standard form, and factor the quadratic equation to find the solutions for the variable. With practice and patience, you can become proficient in solving rational equations and apply them to real-world problems.

Frequently Asked Questions

• Q: What is a rational equation? A: A rational equation is an equation that involves fractions with polynomials in the numerator and denominator.
• Q: How do I solve a rational equation? A: To solve a rational equation, you need to follow these steps: eliminate the fractions by finding the LCM of the denominators, rearrange the equation to get a quadratic equation in standard form, and factor the quadratic equation to find the solutions for the variable.
• Q: What is the least common multiple (LCM)? A: The least common multiple (LCM) of two or more expressions is the smallest expression that is a multiple of each of the given expressions.
• Q: How do I find the LCM of two expressions? A: To find the LCM of two expressions, you can list the multiples of each expression and find the smallest multiple that is common to both.

Real-World Applications

Rational equations have many real-world applications, including:

• Physics: Rational equations are used to describe the motion of objects under the influence of forces.
• Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Rational equations are used to model economic systems and make predictions about future trends.

Conclusion

Solving rational equations can be a challenging task, but with the right approach, it can be done efficiently. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving rational equations and apply them to real-world problems. Remember to eliminate fractions, rearrange the equation to get a quadratic equation in standard form, and factor the quadratic equation to find the solutions for the variable. With practice and patience, you can become proficient in solving rational equations and apply them to real-world problems.