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Linear equations form the backbone of algebra. They involve expressions where the variable, commonly denoted as \( x \), is raised to the power of one, making them the simplest type of algebraic expressions. In any linear equation, you will typically find a structure similar to \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. The main goal when dealing with linear equations is to find the value of the unknown variable that makes the equation true.
Solving linear equations involves isolating the variable on one side of the equation. This means performing operations like addition, subtraction, multiplication, or division. Importantly, whatever operation you perform on one side of the equation must also be done to the other side. Hence, keeping the equation balanced is crucial to finding a correct solution.

After solving an equation, it's always vital to check your solution. This step verifies whether your calculated value for the variable actually satisfies the original equation.
To check a solution, substitute the value you found back into the original equation's variable. For example, if you found \( x = 9 \), plug it into the equation and work through it. If both sides are equal, your solution is verified.
But occasionally, as shown with the exercise, mistakes in the solution process could indicate an error. A detailed re-check of substituting a proposed solution not matching highlights these mishaps and helps correct them before reaching a final answer. This safeguard ensures that the right answer is confirmed.

No matter how good we are, human errors in arithmetic can easily occur, especially when dealing with multiple steps in an equation. Even a tiny miscalculation can lead to an incorrect solution, as demonstrated when re-checking the solution for the given equation. When calculations are off by even the smallest amount, it can lead to the entire solution being incorrect.
Common areas where arithmetic errors may happen include:

• Forgetfulness in carrying over numbers during addition or subtraction
• Misplacing decimal points
• Incorrect execution of multiplication or division calculations

To minimize such errors, redoing your arithmetic or using a calculator for verification can be highly beneficial. This not only helps spot errors but also reinforces confidence in the solution presented.

In solving equations, algebraic manipulation is a key tool. It involves rearranging and simplifying an equation to isolate the variable. The main idea is to perform the opposite operation to both sides of the equation.
For example, if you have the equation \( 3.6x = 2.9x + 6.3 \), you need to gather like terms on one side, as done in the first step of our given solution, obtaining \( 0.7x = 6.3 \). Here, subtraction is used to isolate \( x \) by subtracting \( 2.9x \) from both sides.
Such manipulations allow us to clean the equation up into a simple form where the solution becomes evident. Remember, practicing these algebraic processes can greatly enhance problem-solving skills and make dealing with more complex equations in the future a lot more approachable.