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To solve any linear equation, our first aim is to isolate the variable we're solving for, usually denoted as 'x'. This means we want to get 'x' (or any variable) by itself on one side of the equation. In our example, starting from the equation:
9.1x + 5 = 8.1x

we isolate 'x' by eliminating other terms involving 'x' from one side. We do this by subtracting 8.1x from both sides of the equation. This step ensures that we gather all x terms together. Therefore, we get:

9.1x - 8.1x + 5 = 8.1x - 8.1x

After simplifying, it becomes:

1x + 5 = 0.

By isolating the variable, we mean positioning it alone so that we can eventually solve for its value. This step is crucial in finding the solution.

Once you've found a solution to an equation, it is always good practice to check it. Here, we found that x = -5. To verify our solution, we substitute x back into the original equation:

9.1x + 5 = 8.1x

Substituting x = -5, we get:

9.1(-5) + 5 = 8.1(-5).

• Calculate 9.1(-5) which gives -45.5
• Add 5 to -45.5 to get -40.5

On the right side:

• Calculate 8.1(-5), resulting in -40.5

Both sides are equal (-40.5 = -40.5), thus confirming our solution is correct. Checking your solutions avoids simple errors and assures that your answer is precise.

Understanding basic algebra is essential for solving equations efficiently. Key algebraic principles include:

• Combining like terms: Terms that contain the same variable can be combined, as in 9.1x and -8.1x.
• Maintaining balance: What you do to one side of the equation, you must do to the other side to keep it balanced, ensuring the equation stays true.
• Inverse operations: Use operations that 'undo' each other to solve for the variable. For example, add and subtract or multiply and divide.

Our example shows these principles in action. By subtracting 8.1x and later subtracting 5, we used inverse operations to isolate and solve for 'x'. These fundamental ideas form the foundation of algebra and are critical for progressing in more advanced math topics.