91Ó°ÊÓ

In any linear equation, the first step you need to take is to isolate the variable. This principle is like uncovering a hidden treasure. By isolating the variable, you make the equation simpler and more straightforward. Essentially, isolation means manipulating the equation to get the variable on its own on one side. For example, in the equation \( z+1.9=12.41 \), we want to isolate \( z \). To do this, subtract 1.9 from both sides. This gives us:

\( z + 1.9 - 1.9 = 12.41 - 1.9 \).

You see, we subtracted 1.9 from both sides so that \( z \) is left on its own on the left side of the equation. When working with equations, always remember to perform the same operation on both sides to maintain equality.

Simplifying equations is a crucial step after isolating the variable. Once you've isolated the variable, you'll often have some arithmetic left to do. In our example, we subtracted 1.9 from 12.41, which simplifies to \( 10.51 \). So, we see:

\( z = 12.41 - 1.9 \),

which simplifies to:
\( z = 10.51 \).

Simplification not only makes the equation easier to handle but also gives us the final value of the variable. Whenever you simplify an equation, double-check your arithmetic to avoid mistakes.

Verifying the solution is the final, yet equally important step. It's like double-checking your answer to make sure everything adds up correctly. Once you've isolated the variable and simplified the equation, you should substitute your solution back into the original equation. For our example, we calculated that \( z = 10.51 \). Substituting this back into the original equation: \( 10.51 + 1.9 \) should equal 12.41. Let's check:
\( 10.51+1.9 = 12.41 \).

Since both sides of the equation are equal, our solution is verified. Always take a moment to verify your solutions to confirm their accuracy.