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Isolating the variable is crucial for solving any linear equation. It means getting the variable 'x' (or any other letter) by itself on one side of the equation. Think of it as gathering all the 'x' items together and pushing everything else to the other side. In the example provided: $$ 4x - 7 = 0 $$To isolate the variable term (\text{4x}), you need to move the constant (\text{-7}) to the other side. To do this, you perform the opposite operation of subtraction, which is addition. So, you add \text{7} to both sides:$$ 4x - 7 + 7 = 0 + 7 $$. Simplify to get:$$ 4x = 7 $$. Here, the term with 'x' is now alone on one side. Make sure you do the same operation on both sides of the equation to maintain balance. This is the essence of isolating the variable.

Once you've isolated the variable term, the next step is to solve for 'x'. This step follows isolating the variable and gets you to the solution. From the previous step, we're left with the simplified equation:$$ 4x = 7 $$To solve for 'x', you need to get 'x' by itself. Currently, 'x' is multiplied by 4. To undo multiplication, you do the opposite operation, which is division. Divide both sides by 4:$$ \frac{4x}{4} = \frac{7}{4} $$Simplify this to get:$$ x = \frac{7}{4} $$Now 'x' is by itself, and you've solved for 'x'. Remember, whatever you do to one side, you must do to the other to keep the equation balanced. By dividing both sides by 4, you maintain this balance and find 'x'.

Simplifying equations is about making the equation easier to solve. This often involves combining like terms, removing parentheses, and performing basic arithmetic operations. For our example, we didn't have extra steps in simplification because our terms were already in a form we could work with. However, it's a good idea to understand this concept.Here are steps to simplify equations:

• Combine like terms: If you had terms like \text{3x + 2x}, you would combine them to get \text{5x}.
• Remove parentheses: Use the distributive property to remove parentheses. For example, \text{2(x + 3)} becomes \text{2x + 6}.
• Perform basic operations: Add, subtract, multiply, or divide to reduce the equation to a simpler form.

In our solved equation:$$ 4x - 7 = 0 $$we already had a simplified equation to work with. We added 7 to both sides and then divided by 4 to reach the final answer. Simplifying equations makes it easier to isolate the variable and eventually solve for 'x'.