Homework 2.6: Quadratic SystemsQuestion 12 Solve The Rational Equation: 3 M M 2 − 49 + 8 M − 7 = 1 M + 7 \frac{3m}{m^2-49} + \frac{8}{m-7} = \frac{1}{m+7} M 2 − 49 3 M ​ + M − 7 8 ​ = M + 7 1 ​ Answer: M = M = M = □ \square □ Question 13

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Introduction

Rational equations are a type of algebraic equation that involves fractions with polynomials in both the numerator and denominator. Solving rational equations can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will focus on solving the rational equation 3mm2−49+8m−7=1m+7\frac{3m}{m^2-49} + \frac{8}{m-7} = \frac{1}{m+7}.

Understanding the Equation

Before we dive into solving the equation, let's first understand its components. The equation is a rational equation, which means it involves fractions with polynomials in both the numerator and denominator. The equation is:

3mm2−49+8m−7=1m+7\frac{3m}{m^2-49} + \frac{8}{m-7} = \frac{1}{m+7}

Step 1: Factor the Denominator

The first step in solving this equation is to factor the denominator of the first fraction. The denominator is m2−49m^2-49, which can be factored as (m−7)(m+7)(m-7)(m+7). This gives us:

3m(m−7)(m+7)+8m−7=1m+7\frac{3m}{(m-7)(m+7)} + \frac{8}{m-7} = \frac{1}{m+7}

Step 2: Multiply Both Sides by the Least Common Multiple (LCM)

The next step is to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is (m−7)(m+7)(m-7)(m+7). Multiplying both sides by the LCM gives us:

3m+8(m+7)=(m+7)3m + 8(m+7) = (m+7)

Step 3: Simplify the Equation

Now that we have multiplied both sides by the LCM, we can simplify the equation. Expanding the left-hand side of the equation gives us:

3m+8m+56=m+73m + 8m + 56 = m + 7

Step 4: Combine Like Terms

The next step is to combine like terms on the left-hand side of the equation. Combining like terms gives us:

11m+56=m+711m + 56 = m + 7

Step 5: Subtract m from Both Sides

Subtracting mm from both sides of the equation gives us:

10m+56=710m + 56 = 7

Step 6: Subtract 56 from Both Sides

Subtracting 56 from both sides of the equation gives us:

10m=−4910m = -49

Step 7: Divide Both Sides by 10

Finally, dividing both sides of the equation by 10 gives us:

m=−4910m = -\frac{49}{10}

Conclusion

Solving rational equations can be a challenging task, but with the right approach, it can be done efficiently. In this article, we have solved the rational equation 3mm2−49+8m−7=1m+7\frac{3m}{m^2-49} + \frac{8}{m-7} = \frac{1}{m+7}. The solution to the equation is m=−4910m = -\frac{49}{10}.

Tips and Tricks

  • When solving rational equations, it is essential to factor the denominator and multiply both sides by the least common multiple (LCM) of the denominators.
  • When simplifying the equation, combine like terms and subtract the same value from both sides of the equation.
  • When dividing both sides of the equation by a value, make sure to check if the value is zero to avoid division by zero.

Real-World Applications

Rational equations have numerous real-world applications in various fields, including:

  • Physics: Rational equations are used to describe the motion of objects and the behavior of physical systems.
  • 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 economic trends.

Conclusion

Q: What is a rational equation?

A: A rational equation is a type of algebraic equation that involves fractions with polynomials in both 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. Factor the denominator.
  2. Multiply both sides by the least common multiple (LCM) of the denominators.
  3. Simplify the equation by combining like terms.
  4. Subtract the same value from both sides of the equation.
  5. Divide both sides by a value.

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

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.

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

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that they have in common.

Q: What is the difference between a rational equation and a rational expression?

A: A rational equation is an equation that involves fractions with polynomials in both the numerator and denominator, while a rational expression is an expression that involves fractions with polynomials in both the numerator and denominator.

Q: Can I use a calculator to solve rational equations?

A: Yes, you can use a calculator to solve rational equations, but it's essential to understand the steps involved in solving the equation.

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

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

  • Not factoring the denominator.
  • Not multiplying both sides by the least common multiple (LCM).
  • Not simplifying the equation by combining like terms.
  • Not subtracting the same value from both sides of the equation.
  • Not dividing both sides by a value.

Q: How do I check my answer when solving a rational equation?

A: To check your answer when solving a rational equation, you can plug the solution back into the original equation and simplify it. If the solution is correct, the equation should be true.

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

A: Rational equations have numerous real-world applications in various fields, including:

  • Physics: Rational equations are used to describe the motion of objects and the behavior of physical systems.
  • 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 economic trends.

Q: Can I use rational equations to solve problems in other areas of mathematics?

A: Yes, you can use rational equations to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.

Conclusion

In conclusion, solving rational equations requires a step-by-step approach, including factoring the denominator, multiplying both sides by the least common multiple (LCM), simplifying the equation, and dividing both sides by a value. With practice and patience, solving rational equations can become a breeze.