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Isolating the variable is a fundamental step in solving linear equations. It involves manipulating the equation in a way that places the unknown variable on one side and all other terms on the opposite side. In the process of isolation, you make use of inverse operations to undo the actions that are being applied to the variable.

For example, consider the equation from the exercise, which is \(15 \cdot a = 45\). Here, \(a\) is multiplied by 15. To isolate \(a\), we need to reverse this operation by performing the opposite - which is division. We divide both sides of the equation by 15, that is:

\[\frac{15 \cdot a}{15} = \frac{45}{15}\]

This mathematical step removes the factor of 15 from the left-hand side, leaving \(a\) by itself, hence 'isolated'. The equation has been simplified to the point where the solution for \(a\) is now clear.

Equation simplification is reducing an equation to its simplest form, making it easier to solve or understand. This process usually involves combining like terms, reducing fractions, and performing arithmetic operations.

Combining Like Terms

When simplifying an equation, any similar terms on the same side of the equation can be combined into a single term for clarity.

Reducing Fractions

If any fractions are present, simplifying them to the lowest terms can also make the equation easier to work with. In our example, after dividing both sides by 15, we obtain a simple fraction \(\frac{45}{15}\) which simplifies to 3. Thus, the equation now reads \(a = 3\), which is the simplest form.

After finding a solution to an equation, it is crucial to ensure its validity by checking the math solution. This means substituting the obtained value back into the original equation to verify if it maintains equality.

In our textbook problem, the solution \(a = 3\) can be checked by substituting \(a\) with 3 in the original equation \(15 \cdot a = 45\). This results in \(15 \cdot 3 = 45\), which simplifies to \(45 = 45\). Since both sides of the equation are equal, the solution \(a = 3\) is confirmed to be correct.

• This check acts as a guard against common calculation mistakes.
• Always practice this habit to ensure that you haven't overlooked any errors in your algebraic manipulation.
• It reinforces your understanding and confidence in the solution you've produced.