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Isolating variables is a fundamental concept in solving linear equations. The goal is to get the variable alone on one side of the equation.
When you have an equation like \(26 + 13d = 14d + 11\), you want to isolate \(d\).
First, you need to move all the terms with the variable to one side of the equation. In Step 1, we subtracted \(13d\) from both sides: \(26 + 13d - 13d = 14d + 11 - 13d\).
This simplified to \(26 = d + 11\). Now, \(d\) is on one side of the equation, but it's not fully isolated yet.
We then subtract 11 from both sides in Step 2: \(26 - 11 = d + 11 - 11\), simplifying to \(15 = d\). By doing this, we've successfully isolated \(d\).
Remember to always perform the same operation on both sides to maintain balance, which brings us to our next point.

Balancing equations is crucial when solving for a variable. To keep an equation balanced, whatever you do to one side, you must do to the other.
In Step 1 of our example equation, we subtracted \(13d\) from both sides: \(26 + 13d - 13d = 14d + 11 - 13d\). This operation keeps the equation balanced.
Similarly, in Step 2, subtracting 11 from both sides \(26 - 11 = d + 11 - 11\), keeps things balanced too.
Here are a few tips for balancing equations:

• Always perform the same mathematical operation on both sides.
• Check your work by verifying that both sides of the equation are still equal after each operation.
• If you multiply or divide by a number, ensure you apply this to every term on both sides.

Keeping an equation balanced ensures that the solution you find for the variable is accurate.

Understanding the difference between constants and variables is essential in algebra.
A variable is a symbol, usually a letter, that represents a number we don't know yet. In our example, \(d\) is the variable.
On the other hand, constants are fixed values. In our problem, 26 and 11 are constants.
To solve equations, you need to manipulate these constants and variables. In our example \(26 + 13d = 14d + 11\), the 26 and 11 are constants, while 13d and 14d contain the variable \(d\).
We isolated \(d\) by moving the terms with the variable to one side and constants to the other.

• Recognizing which terms are constants and which are variables helps you know what to move and how.
• Perform operations that simplify the equation by making the variable easy to isolate.
• Remember that variables can change, their exact value depends on the equation they're in.

In summary, distinguishing between constants and variables enables you to correctly manipulate the equations to find solutions.