For A Given Input Value \[$ M \$\], The Function \[$ F \$\] Outputs A Value \[$ N \$\] To Satisfy The Following Equation:$\[ 7m + 2 = 6n - 5 \\]Write A Formula For \[$ F(m) \$\] In Terms Of \[$ M \$\].

Introduction

In mathematics, solving equations is a fundamental concept that helps us understand the relationship between variables. Given an equation, we can use various techniques to isolate the variable and find its value. In this article, we will focus on solving the equation $7m + 2 = 6n - 5$ to find a formula for $f(m)$ in terms of $m$.

Understanding the Equation

The given equation is $7m + 2 = 6n - 5$. Our goal is to express $f(m)$ in terms of $m$, which means we need to isolate $n$ in terms of $m$. To do this, we can use algebraic manipulations to simplify the equation.

Isolating n

To isolate $n$, we can start by adding $5$ to both sides of the equation:

$$7m + 2 + 5 = 6n - 5 + 5$$

This simplifies to:

$$7m + 7 = 6n$$

Next, we can subtract $7$ from both sides of the equation:

$$7m = 6n - 7$$

Expressing n in Terms of m

Now, we can express $n$ in terms of $m$ by dividing both sides of the equation by $6$:

$$\frac{7m}{6} = n - \frac{7}{6}$$

To isolate $n$, we can add $\frac{7}{6}$ to both sides of the equation:

$$n = \frac{7m}{6} + \frac{7}{6}$$

Finding the Formula for f(m)

Since $f(m)$ outputs a value $n$ to satisfy the equation, we can express $f(m)$ as:

$$f(m) = \frac{7m}{6} + \frac{7}{6}$$

This is the formula for $f(m)$ in terms of $m$.

Simplifying the Formula

We can simplify the formula by combining the fractions:

$$f(m) = \frac{7m + 7}{6}$$

Conclusion

In this article, we solved the equation $7m + 2 = 6n - 5$ to find a formula for $f(m)$ in terms of $m$. We used algebraic manipulations to isolate $n$ in terms of $m$ and then expressed $f(m)$ as a function of $m$. The final formula for $f(m)$ is $\frac{7m + 7}{6}$.

Example Use Case

Suppose we want to find the value of $f(3)$. We can plug in $m = 3$ into the formula:

$$f(3) = \frac{7(3) + 7}{6}$$

Simplifying the expression, we get:

$$f(3) = \frac{21 + 7}{6} = \frac{28}{6} = \frac{14}{3}$$

Therefore, the value of $f(3)$ is $\frac{14}{3}$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Start with the given equation: $7m + 2 = 6n - 5$
2. Add $5$ to both sides of the equation: $7m + 7 = 6n$
3. Subtract $7$ from both sides of the equation: $7m = 6n - 7$
4. Divide both sides of the equation by $6$: $\frac{7m}{6} = n - \frac{7}{6}$
5. Add $\frac{7}{6}$ to both sides of the equation: $n = \frac{7m}{6} + \frac{7}{6}$
6. Express $f(m)$ as: $f(m) = \frac{7m}{6} + \frac{7}{6}$
7. Simplify the formula: $f(m) = \frac{7m + 7}{6}$

Key Takeaways

• To solve the equation $7m + 2 = 6n - 5$, we need to isolate $n$ in terms of $m$.
• We can use algebraic manipulations to simplify the equation and express $n$ in terms of $m$.
• The final formula for $f(m)$ is $\frac{7m + 7}{6}$.
• We can use the formula to find the value of $f(m)$ for any given value of $m$.
  

Q: What is the main goal of solving the equation $7m + 2 = 6n - 5$?

A: The main goal is to express $f(m)$ in terms of $m$, which means we need to isolate $n$ in terms of $m$.

Q: How do we start solving the equation?

A: We start by adding $5$ to both sides of the equation to simplify it.

Q: What is the next step after adding $5$ to both sides of the equation?

A: We subtract $7$ from both sides of the equation to isolate $n$.

Q: How do we express $n$ in terms of $m$?

A: We divide both sides of the equation by $6$ to express $n$ in terms of $m$.

Q: What is the final formula for $f(m)$?

A: The final formula for $f(m)$ is $\frac{7m + 7}{6}$.

Q: Can we simplify the formula further?

A: Yes, we can simplify the formula by combining the fractions.

Q: How do we use the formula to find the value of $f(m)$ for any given value of $m$?

A: We can plug in the value of $m$ into the formula and simplify the expression to find the value of $f(m)$.

Q: What is the value of $f(3)$?

A: The value of $f(3)$ is $\frac{14}{3}$.

Q: Can you provide a step-by-step solution to the problem?

A: Yes, here is a step-by-step solution:

1. Start with the given equation: $7m + 2 = 6n - 5$
2. Add $5$ to both sides of the equation: $7m + 7 = 6n$
3. Subtract $7$ from both sides of the equation: $7m = 6n - 7$
4. Divide both sides of the equation by $6$: $\frac{7m}{6} = n - \frac{7}{6}$
5. Add $\frac{7}{6}$ to both sides of the equation: $n = \frac{7m}{6} + \frac{7}{6}$
6. Express $f(m)$ as: $f(m) = \frac{7m}{6} + \frac{7}{6}$
7. Simplify the formula: $f(m) = \frac{7m + 7}{6}$

Q: What are the key takeaways from solving the equation?

A: The key takeaways are:

• To solve the equation $7m + 2 = 6n - 5$, we need to isolate $n$ in terms of $m$.
• We can use algebraic manipulations to simplify the equation and express $n$ in terms of $m$.
• The final formula for $f(m)$ is $\frac{7m + 7}{6}$.
• We can use the formula to find the value of $f(m)$ for any given value of $m$.

Q: Can you provide an example use case for the formula?

A: Yes, suppose we want to find the value of $f(4)$. We can plug in $m = 4$ into the formula:

$$f(4) = \frac{7(4) + 7}{6}$$

Simplifying the expression, we get:

$$f(4) = \frac{28 + 7}{6} = \frac{35}{6}$$

Therefore, the value of $f(4)$ is $\frac{35}{6}$.

Q: Can you provide a real-world application of the formula?

A: Yes, the formula can be used in various real-world applications, such as:

• Calculating the cost of goods sold in a business
• Determining the amount of material needed for a construction project
• Finding the value of a variable in a scientific experiment

The formula can be used in any situation where we need to express a value in terms of a variable.