Solve M M + 6 + 8 M + 2 = 1 \frac{m}{m+6}+\frac{8}{m+2}=1 M + 6 M ​ + M + 2 8 ​ = 1 Your Final Line Should Say, M = … M=\ldots M = …

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Introduction

Solving equations involving fractions can be a challenging task, especially when the fractions have different denominators. In this article, we will focus on solving the equation $\frac{m}{m+6}+\frac{8}{m+2}=1$. This equation involves two fractions with different denominators, and we will use various techniques to simplify and solve it.

Understanding the Equation

The given equation is $\frac{m}{m+6}+\frac{8}{m+2}=1$. To solve this equation, we need to first understand the concept of equivalent fractions. Equivalent fractions are fractions that have the same value, but with different numerators and denominators. In this case, we can rewrite the fractions as follows:

$\frac{m}{m+6}=\frac{m}{m+6}\cdot\frac{m+2}{m+2}=\frac{m(m+2)}{(m+6)(m+2)}$

$\frac{8}{m+2}=\frac{8}{m+2}\cdot\frac{m+6}{m+6}=\frac{8(m+6)}{(m+2)(m+6)}$

Simplifying the Equation

Now that we have rewritten the fractions, we can simplify the equation by combining the fractions on the left-hand side. To do this, we need to find a common denominator for the two fractions. The common denominator is $(m+6)(m+2)$.

$\frac{m(m+2)}{(m+6)(m+2)}+\frac{8(m+6)}{(m+2)(m+6)}=1$

Combining the Fractions

Now that we have a common denominator, we can combine the fractions by adding the numerators.

$\frac{m(m+2)+8(m+6)}{(m+6)(m+2)}=1$

Expanding the Numerator

To simplify the numerator, we can expand it by multiplying the terms.

$m(m+2)+8(m+6)=m^2+2m+8m+48$

Simplifying the Numerator

Now that we have expanded the numerator, we can simplify it by combining like terms.

$m^2+2m+8m+48=m^2+10m+48$

Substituting the Simplified Numerator

Now that we have simplified the numerator, we can substitute it back into the equation.

$\frac{m^2+10m+48}{(m+6)(m+2)}=1$

Cross-Multiplying

To eliminate the fraction, we can cross-multiply by multiplying both sides of the equation by the denominator.

$m^2+10m+48=(m+6)(m+2)$

Expanding the Right-Hand Side

To simplify the right-hand side, we can expand it by multiplying the terms.

$(m+6)(m+2)=m^2+2m+6m+12$

Simplifying the Right-Hand Side

Now that we have expanded the right-hand side, we can simplify it by combining like terms.

$m^2+2m+6m+12=m^2+8m+12$

Equating the Numerators

Now that we have simplified both sides of the equation, we can equate the numerators.

$m^2+10m+48=m^2+8m+12$

Subtracting the Common Term

To eliminate the common term on both sides of the equation, we can subtract it from both sides.

$m^2+10m+48-m^2-8m-12=0$

Simplifying the Equation

Now that we have subtracted the common term, we can simplify the equation by combining like terms.

$2m+36=0$

Solving for m

To solve for m, we can isolate the variable by subtracting 36 from both sides of the equation.

$2m=-36$

Dividing Both Sides

To solve for m, we can divide both sides of the equation by 2.

$m=\frac{-36}{2}$

Simplifying the Expression

Now that we have divided both sides of the equation by 2, we can simplify the expression by evaluating the fraction.

$m=-18$

The final answer is $m=\boxed{-18}$.

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Introduction

In our previous article, we solved the equation $\frac{m}{m+6}+\frac{8}{m+2}=1$ and found that the solution is $m=-18$. However, we understand that some readers may still have questions about the solution process. In this article, we will address some of the most frequently asked questions about solving the equation.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to rewrite the fractions with a common denominator. This involves multiplying the numerator and denominator of each fraction by the denominator of the other fraction.

Q: Why do we need to find a common denominator?

A: We need to find a common denominator to combine the fractions on the left-hand side of the equation. This allows us to add the numerators and simplify the equation.

Q: How do we simplify the numerator?

A: To simplify the numerator, we can expand it by multiplying the terms. This involves multiplying each term in the numerator by the other terms.

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

A: After simplifying the numerator, we can substitute it back into the equation. This involves replacing the original numerator with the simplified one.

Q: How do we eliminate the fraction?

A: To eliminate the fraction, we can cross-multiply by multiplying both sides of the equation by the denominator. This involves multiplying both sides of the equation by the common denominator.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to solve for the variable. This involves isolating the variable by subtracting the common term from both sides of the equation and then dividing both sides by the coefficient of the variable.

Q: What is the solution to the equation?

A: The solution to the equation is $m=-18$.

Q: Why is the solution $m=-18$?

A: The solution $m=-18$ is obtained by solving the equation $2m+36=0$. This involves subtracting 36 from both sides of the equation and then dividing both sides by 2.

Q: Can you provide an example of how to solve the equation?

A: Yes, we can provide an example of how to solve the equation. Let's say we want to solve the equation $\frac{x}{x+3}+\frac{5}{x+1}=2$. We can follow the same steps as before to solve the equation.

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

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

• Not finding a common denominator
• Not simplifying the numerator
• Not eliminating the fraction
• Not solving for the variable
• Not checking the solution

Q: How can I practice solving equations like this?

A: You can practice solving equations like this by working through examples and exercises. You can also try solving different types of equations, such as linear equations and quadratic equations.

Q: Where can I find more resources on solving equations?

A: You can find more resources on solving equations by checking out online resources, such as Khan Academy and Mathway. You can also consult a math textbook or seek help from a math tutor.

The final answer is $m=\boxed{-18}$.