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Isolating the variable is the first important step in solving linear equations. The goal is to get the unknown variable on its own on one side of the equation. This involves using basic arithmetic operations such as addition, subtraction, multiplication, or division. By isolating the variable, it becomes easier to solve for its value.

In the given problem, we started with the equation: \(5 = z + 19\). To isolate \(z\), we needed to remove the 19 from the right-hand side of the equation. We did this by subtracting 19 from both sides. This keeps the balance of the equation intact. The equation now looks like this: \[\begin{aligned} 5 - 19 &= z\end{aligned}\]. By performing the subtraction, we get closer to finding the value of the variable.

Simplifying the equation is the next step after isolating the variable. This usually involves performing arithmetic operations to simplify expressions or terms on either side of the equation.

In our example: \(5 - 19 = z\). Now, we need to simplify the left-hand side. First, calculate the result of the subtraction: \(5 - 19\) equals \(-14\), so the simplified equation is: \(-14 = z\). Therefore, the value of \(z\) is \(z = -14\). This gives us the simplified and final form of the equation.

It is always important to check your solutions to ensure they are correct. After finding the value of the variable, substitute it back into the original equation to verify the result.

For the equation: \(5 = z + 19\), we found \(z = -14\). Now, substitute \(-14\) back into the original equation: \[5 = -14 + 19\]. Simplify the right side: \[-14 + 19 = 5\]. Both sides of the equation are equal: \[5 = 5\]. Since both sides match, our solution \(z = -14\) is verified to be correct. This step ensures that the process of solving the equation was executed properly and the solution obtained is accurate.