91Ó°ÊÓ

Isolating the variable is our primary goal when solving equations with variables on both sides. This means we need to make sure all terms containing the variable are on one side of the equation, and constant terms are on the other side. In the exercise, we start with the equation \(21k = 20k - 11\). Here, \(21k\) and \(20k\) are our variable terms.
To isolate the variable term, we'll subtract \(20k\) from both sides of the equation. Doing the same operation on both sides keeps the equation balanced. After this step, the equation looks like this:
\[21k - 20k = 20k - 11 - 20k\]This simplifies to:
\[k = -11 \]Now, the variable terms are isolated, and we see the value of \(k\) by itself.

Two-step equations usually involve an initial step to simplify or rearrange the terms before isolating the variable. In this exercise, we dealt with a simple equation: \(21k = 20k - 11\). Our first step was to get all variable terms on one side by subtracting \(20k\). This transformation simplified it directly to the solution.
However, not all equations will simplify this quickly. Generally, you'll need to:

• Combine like terms on each side of the equation.
• Move variable terms to one side (adding/subtracting terms).
• Move constant terms to the other side.
• Finally, isolate the variable.

By following these steps, many algebraic equations simplify into manageable two-step processes.

Once you've isolated the variable and solved the equation, it’s essential to verify your solution. This ensures that no mistakes were made during the solving process. To do this, simply substitute the found value back into the original equation. Let's verify our solution for \(k\) in the exercise:
Our solution was \(k = -11\). Substituting this into the original equation, we check:
\[21(-11) = 20(-11) - 11\]Simplifying both sides, we get:
\[-231 = -220 - 11\]Both sides are equal, confirming that our solution, \(k = -11\), is correct. Verification is a good practice to catch errors and ensures accuracy in your solutions.