Linear equations are one of the fundamental concepts in mathematics, particularly in algebra. A linear equation is an equation that a straight line on a graph can represent, and it contains one or more variables raised to the first power. In a linear equation, the highest power of the variable is always 1, and there is only one solution.

To solve a linear equation, we need to find the variable's value that makes the equation true. To do this, we must manipulate the equation by adding, subtracting, multiplying, or dividing both sides by the same number. We must keep the equation balanced at all times, meaning that whatever we do to one side of the equation, we must do to the other.

Consider the following example: 3x + 4 = 7x - 1. We aim to find the value of x that makes this equation true. One way to solve the equation is to simplify each side of this equation by combining the like terms. We can start by subtracting 3x from both sides of this equation to get rid of the variable on one side:

3x + 4 - 3x = 7x - 1 - 3x

Simplifying, we get:
4 = 4x - 1

Next, we can add 1 to both sides of this equation to isolate the term with the variable on one side:

4 + 1 = 4x - 1 + 1

Simplifying, we get:
5 = 4x

Finally, we can divide both sides of this equation by 4 to solve for x:
$$\frac{5}{4}=x$$

So, $$x=\frac{5}{4}$$

Another method for solving linear equations is the "cover-up" method mentioned in the prompt. This method involves covering up the term with the variable on one side of this equation and then solving for the variable using basic arithmetic operations.