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A Step-by-Step Guide to Solving the Given Equation

Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the various techniques involved in solving different types of equations. In this article, we will focus on solving a rational equation involving a variable $w$. The given equation is:

$$\frac{w}{w+4}+\frac{3}{w+5}=\frac{w+2}{w^2+9w+20}$$

Our goal is to solve for $w$ and find the values that satisfy the equation.

Step 1: Factor the Denominator

The first step in solving this equation is to factor the denominator of the right-hand side. We can factor the quadratic expression $w^2+9w+20$ as follows:

$$w^2+9w+20 = (w+4)(w+5)$$

Now, the equation becomes:

$$\frac{w}{w+4}+\frac{3}{w+5}=\frac{w+2}{(w+4)(w+5)}$$

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

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is $(w+4)(w+5)$.

$$\frac{w}{w+4}+\frac{3}{w+5}=\frac{w+2}{(w+4)(w+5)}$$

$$\frac{w(w+4)(w+5)+3(w+4)(w+5)}{(w+4)(w+5)}=\frac{(w+2)(w+4)(w+5)}{(w+4)(w+5)}$$

Step 3: Simplify the Equation

Now, we can simplify the equation by canceling out the common factors:

$$w(w+4)(w+5)+3(w+4)(w+5)=(w+2)(w+4)(w+5)$$

Expanding the left-hand side, we get:

$$w^3+9w^2+20w+3w^2+36w+60=w^3+9w^2+20w+2w^2+8w+10w+20$$

Combining like terms, we get:

$$w^3+12w^2+56w+60=w^3+11w^2+26w+20$$

Step 4: Subtract $w^3+11w^2+26w+20$ from Both Sides

Subtracting $w^3+11w^2+26w+20$ from both sides, we get:

$$w^3+12w^2+56w+60-(w^3+11w^2+26w+20)=0$$

Simplifying, we get:

$$w^2+30w+40=0$$

Step 5: Factor the Quadratic Expression

The quadratic expression $w^2+30w+40$ can be factored as follows:

$$w^2+30w+40 = (w+10)(w+4)$$

Now, the equation becomes:

$$(w+10)(w+4)=0$$

Step 6: Solve for $w$

To solve for $w$, we can set each factor equal to zero and solve for $w$:

$$w+10=0 \Rightarrow w=-10$$

$$w+4=0 \Rightarrow w=-4$$

Conclusion

In this article, we solved a rational equation involving a variable $w$. We started by factoring the denominator, then multiplied both sides by the least common multiple (LCM), simplified the equation, and finally factored the quadratic expression. We found two solutions for $w$: $w=-10$ and $w=-4$.

Discussion

The given equation is a rational equation, and solving it requires careful manipulation of the fractions. The key steps involved in solving this equation are factoring the denominator, multiplying both sides by the least common multiple (LCM), simplifying the equation, and factoring the quadratic expression. By following these steps, we can solve rational equations involving variables.

Final Answer

The final answer is: $\boxed{-10, -4}$

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Frequently Asked Questions

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are expressions that can be written in the form of a fraction, where the numerator and denominator are polynomials.

Q: How do I solve a rational equation?

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

1. Factor the denominator of the equation.
2. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
3. Simplify the equation.
4. Factor the quadratic expression, if possible.
5. Solve for the variable.

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

A: The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers. In the context of rational equations, the LCM is the smallest expression that can be divided by each of the denominators.

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

A: To find the LCM of two or more expressions, you need to list the factors of each expression and find the smallest expression that contains all the factors.

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

A: A rational expression is an expression that can be written in the form of a fraction, where the numerator and denominator are polynomials. A rational equation is an equation that contains one or more rational expressions.

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

A: Yes, you can use a calculator to solve a rational equation. However, it's always a good idea to check your work by hand to make sure the solution is correct.

Q: What if I have a rational equation with a variable in the denominator?

A: If you have a rational equation with a variable in the denominator, you need to be careful when multiplying both sides of the equation by the LCM. You may need to use a different method, such as using a conjugate to eliminate the variable from the denominator.

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

A: Yes, you can use the quadratic formula to solve a rational equation. However, you need to make sure that the equation is in the form of a quadratic equation, and that the coefficients are correct.

Q: What if I have a rational equation with a complex solution?

A: If you have a rational equation with a complex solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the complex numbers.

Q: Can I use a graphing calculator to solve a rational equation?

A: Yes, you can use a graphing calculator to solve a rational equation. However, it's always a good idea to check your work by hand to make sure the solution is correct.

Q: What if I have a rational equation with a rational solution?

A: If you have a rational equation with a rational solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the rational numbers.

Q: Can I use a computer algebra system (CAS) to solve a rational equation?

A: Yes, you can use a computer algebra system (CAS) to solve a rational equation. However, it's always a good idea to check your work by hand to make sure the solution is correct.

Q: What if I have a rational equation with a variable in the numerator and denominator?

A: If you have a rational equation with a variable in the numerator and denominator, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the variable from the numerator and denominator.

Q: Can I use a rational equation to model real-world problems?

A: Yes, you can use a rational equation to model real-world problems. For example, you can use a rational equation to model the growth of a population, or the flow of a fluid through a pipe.

Q: What if I have a rational equation with a non-rational solution?

A: If you have a rational equation with a non-rational solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the non-rational numbers.

Q: Can I use a rational equation to solve a system of equations?

A: Yes, you can use a rational equation to solve a system of equations. For example, you can use a rational equation to solve a system of linear equations, or a system of quadratic equations.

Q: What if I have a rational equation with a variable in the denominator and a constant in the numerator?

A: If you have a rational equation with a variable in the denominator and a constant in the numerator, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the variable from the denominator.

Q: Can I use a rational equation to solve a quadratic equation?

A: Yes, you can use a rational equation to solve a quadratic equation. For example, you can use a rational equation to solve a quadratic equation with a variable in the numerator and a constant in the denominator.

Q: What if I have a rational equation with a variable in the numerator and a variable in the denominator?

A: If you have a rational equation with a variable in the numerator and a variable in the denominator, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the variables from the numerator and denominator.

Q: Can I use a rational equation to solve a system of quadratic equations?

A: Yes, you can use a rational equation to solve a system of quadratic equations. For example, you can use a rational equation to solve a system of quadratic equations with variables in the numerator and denominator.

Q: What if I have a rational equation with a non-linear solution?

A: If you have a rational equation with a non-linear solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the non-linear numbers.

Q: Can I use a rational equation to solve a system of non-linear equations?

A: Yes, you can use a rational equation to solve a system of non-linear equations. For example, you can use a rational equation to solve a system of non-linear equations with variables in the numerator and denominator.

Q: What if I have a rational equation with a complex solution and a non-rational solution?

A: If you have a rational equation with a complex solution and a non-rational solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the complex and non-rational numbers.

Q: Can I use a rational equation to solve a system of complex equations?

A: Yes, you can use a rational equation to solve a system of complex equations. For example, you can use a rational equation to solve a system of complex equations with variables in the numerator and denominator.

Q: What if I have a rational equation with a non-linear solution and a complex solution?

A: If you have a rational equation with a non-linear solution and a complex solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the non-linear and complex numbers.

Q: Can I use a rational equation to solve a system of non-linear and complex equations?

A: Yes, you can use a rational equation to solve a system of non-linear and complex equations. For example, you can use a rational equation to solve a system of non-linear and complex equations with variables in the numerator and denominator.

Q: What if I have a rational equation with a variable in the numerator and a variable in the denominator, and a non-linear solution?

A: If you have a rational equation with a variable in the numerator and a variable in the denominator, and a non-linear solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the variables from the numerator and denominator.

Q: Can I use a rational equation to solve a system of equations with variables in the numerator and denominator, and non-linear solutions?

A: Yes, you can use a rational equation to solve a system of equations with variables in the numerator and denominator, and non-linear solutions. For example, you can use a rational equation to solve a system of equations with variables in the numerator and denominator, and non-linear solutions with variables in the numerator and denominator.

Q: What if I have a rational equation with a variable in the numerator and a variable in the denominator, and a complex solution?

A: If you have a rational equation with a variable in the numerator and a variable in the denominator, and a complex solution, you need to be careful when simplifying the equation. You may need to use a different method, such as using a conjugate to eliminate the variables from the numerator and denominator.

Q: Can I use a rational equation to solve a system of equations with variables in the numerator and denominator, and complex solutions?

A: Yes, you can use a rational equation to solve a system of equations with variables in the numerator and denominator, and complex solutions. For example, you can use a rational equation to solve a system of equations with variables in the numerator and denominator, and complex solutions with variables in the numerator and denominator.

Q: What if I have a rational equation with a variable in the numerator and a variable in the