How to Solve Linear Equations: Step-by-Step Guide

Solving linear equations is a fundamental skill in mathematics that helps students understand algebraic concepts. Let's walk through the process using the equation $$\frac{2f}{5} - \frac{8}{5} + 2f = 8$$.

### Step 1: Combine Like Terms
Start by combining the terms that involve the variable $$f$$. In this equation, we have two terms with $$f$$: $$\frac{2f}{5}$$ and $$2f$$. To combine them, convert $$2f$$ into a fraction with a denominator of 5:

$$2f = \frac{10f}{5}$$.

Now, we can rewrite the equation as:

$$\frac{2f}{5} + \frac{10f}{5} - \frac{8}{5} = 8$$.

Combine the terms:

$$\frac{2f + 10f}{5} - \frac{8}{5} = 8$$,
which simplifies to:

$$\frac{12f}{5} - \frac{8}{5} = 8$$.

### Step 2: Isolate the Variable
Next, we need to isolate $$\frac{12f}{5}$$. To do this, add $$\frac{8}{5}$$ to both sides of the equation:

$$\frac{12f}{5} = 8 + \frac{8}{5}$$.

To combine the right-hand side, convert the whole number 8 into fifths:

$$8 = \frac{40}{5},$$
so now we have:

$$\frac{12f}{5} = \frac{40}{5} + \frac{8}{5} = \frac{48}{5}.$$

### Step 3: Clear the Fraction
To eliminate the fraction, multiply both sides of the equation by 5:

$$12f = 48.$$

### Step 4: Solve for $$f$$
Finally, divide both sides by 12 to solve for $$f$$:

$$f = \frac{48}{12} = 4.$$

### Conclusion
The correct answer is $$f = 4$$. This example illustrates the importance of combining like terms, isolating the variable, and clearing fractions when solving linear equations. Learning to solve equations not only prepares students for more advanced math topics but also enhances problem-solving skills that can be applied in everyday life, such as budgeting or planning.

By mastering these steps, students develop a strong foundation in algebra that will serve them well in their academic journey and beyond.