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Linear equations are mathematical expressions where the highest exponent of the variable is one. These equations typically take the form of \(ax + b = c\). In our exercise, the equation is \(2.19 = 8 + y\). Here, \(y\) is the variable we are solving for.
Linear equations with one variable usually entail straightforward manipulation to find the variable's value. The main goal is to have the variable on one side of the equation and the numbers on the other side. This is often referred to as putting the equation in its simplest form.
Understanding linear equations is fundamental because they are a building block for more complex algebraic expressions. Also, they have numerous applications in various real-world scenarios, like predicting trends or calculating budgets.

Once you've found a potential solution for an equation, it's essential to verify it. This process ensures accuracy and confirms that the equation balances with the given solution. Let's look at how to check a solution using our example:

• Begin by substituting the solution back into the original equation. Here, the solution for \(y\) is \(-5.81\).
• Replace \(y\) with \(-5.81\) in the original equation, so it becomes \(2.19 = 8 + (-5.81)\).

This step-by-step substitution into the equation allows us to confirm the validity of the calculated solution. If both sides of the equation equal after substitution, our solution is correct. Otherwise, review the problem-solving steps for potential errors.

Variable isolation is a crucial step in solving equations. It involves rearranging the equation so that the variable stands alone on one side. In doing so, you effectively solve for the unknown. This process typically includes performing inverse operations, such as addition or subtraction. In the equation \(2.19 = 8 + y\), we isolated \(y\) by subtracting 8 from both sides. This resulted in the left side being \(-5.81\). Thus, the equation became \(-5.81 = y\), meaning \(y\) was isolated successfully. Ultimately, mastering variable isolation leads to more efficient problem solving. It helps in breaking down complex problems into simpler ones and is a fundamental skill in algebra. Understanding and practicing these steps can greatly enhance one's confidence in tackling various equations.