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The process of solving linear equations involves a series of straightforward steps aimed at finding the value of the variable that makes the equation true. Let's dive into how we tackled the exercise's linear equation:

Step 1: Begin by simplifying the equation if it's not in its simplest form. In the case of our exercise, we're given a clear equation to work with, so this step is already complete.

Step 2: The next goal is to isolate the variable, which means getting the variable by itself on one side of the equation. We can accomplish this by using operations such as addition or subtraction to move terms from one side to the other. For our exercise, we added the term involving the variable 'x' from the right side to the left, balancing the equation and effectively beginning the isolation process.

Step 3: After isolating terms containing the variable, simplify further if necessary. In our exercise, this meant combining like terms to further clear the path toward isolating the variable.

Step 4: Finally, solve for the variable. This usually involves dividing or multiplying both sides of the equation to get the variable alone. In the exercise, we divided both sides by the coefficient of 'x' to find its value.

Step 5: Don't forget the final check! Plugging the solution back into the original equation should yield a true statement. If it does, you've found the correct solution.

To isolate the variable means to rearrange the equation so that the variable stands alone on one side of the equals sign, leaving a coefficient of 1. This allows us to find what the variable equals—its value.

In the exercise we worked with, the steps to isolate the variable 'x' looked like this:

We started by collecting all the terms with 'x' on one side. We then simplified the equation to get a solitary 'x' term. With the 'x' now on one side and the constants on the other, we further simplified by subtraction or addition, leading us to the step where we can divide both sides of the equation to remove the coefficient of 'x'. The result is 'x' isolated and ready for the final solution, which is arrived at by performing the arithmetic.

Isolating the variable is the crux of solving a linear equation, and understanding this process is key to algebraic problem-solving.

Once we have solved for the variable and found a numerical value, we need to check that our solution is correct. This verification step is crucial as it ensures accuracy and understanding of the algebraic process.

To check the solution, simply take the value of the variable and plug it back into the original equation. If both sides of the equation balance out mathematically, then the solution is correct. As we look back at our example exercise, we see that after inserting our found value of 'x' into the equation, we indeed get a balanced equation.

When checking, keep in mind that any rounding or approximation should be taken into consideration, and the equation may not equal perfectly due to these slight variances. If the equation doesn't balance, it could indicate a mistake in the process, prompting a need to retrace steps to find and correct the error.

Checking the mathematical solution reinforces the skills of calculating and validating solutions, making it an essential practice in the world of algebra.