91Ó°ÊÓ

When solving linear equations, one of the main goals is to isolate the variable. This means getting the variable by itself on one side of the equation. In our example, the equation is:

$$ 31 = v - 25 $$

To isolate 'v', we need to get rid of the number that is with 'v'. In this case, it’s -25. We do this by performing the opposite operation. Since -25 is subtracted from 'v', we add 25 to both sides to cancel it out. This step looks like this:

\(31 + 25 = v - 25 + 25 \)

Adding 25 cancels out the -25 on the right-hand side. This leaves us with a simpler equation that only contains the variable on one side. It’s critical to always do the same operation to both sides to keep the equation balanced.

After isolating the variable, the next step is to simplify both sides of the equation. Simplifying means performing the calculations to reduce each side of the equation to its simplest form. Let’s continue with our example:

\(31 + 25 = v\)

We need to add 31 and 25 together to simplify the left-hand side. Performing this addition gives us:

\(56 = v\)
Now the equation is simplified, making it clear that 'v' equals 56. Simplifying helps make the final solution more apparent and easier to understand.

Balancing equations is a crucial concept in solving linear equations. An equation is like a balance scale – both sides must be equal. This means whatever you do to one side, you must do to the other to keep the equation balanced. In our equation:

\(31 = v - 25 \)

We added 25 to both sides to isolate the variable:

\(31 + 25 = v - 25 + 25 \)

After this operation, we got:

As you can see, both sides are balanced every step of the way. Keeping your equation balanced ensures that the variable's value remains correct. This fundamental principle is at the heart of solving any algebraic equation.