91Ó°ÊÓ

Solving equations is a fundamental part of prealgebra. It involves finding the value of the variable that makes an equation true. Each side of the equation expresses the same value, and your task is to manipulate the equation so that you find the correct value for the unknown variable.
Let's consider the given equation: \(-\frac{1}{6}n = -30\). Here, the challenge is to find the value of \(n\) that makes both sides equal.
In solving equations, you can use various techniques such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number. The goal is to isolate the variable on one side, with the outcome set on the other side. Remember, whatever you do on one side, must also be done on the opposing side to maintain balance.

The isolation of a variable is crucial for solving equations. This involves getting the variable alone on one side of the equation. In our example equation, \(-\frac{1}{6}n = -30\), the variable \(n\) is multiplied by \(-\frac{1}{6}\). To isolate \(n\), we need to reverse this operation.
Since \(n\) is being multiplied by \(-\frac{1}{6}\), we can multiply both sides of the equation by \(-6\). This is because multiplying by the reciprocal (in this case, \(-6\)) cancels out the fraction and leaves \(n\) alone.
Thus, we multiply both sides:

• Left side: \(-6 \times -\frac{1}{6}n\) simplifies to \(n\).
• Right side: \(-30 \times -6\) equals \(180\).

This results in \(n = 180\), effectively isolating \(n\). This technique maintains equality while allowing you to solve for the variable easily.

Once you've isolated the variable and found a solution, it's important to verify that the solution is correct. Checking solutions involves substituting the found value back into the original equation to ensure that it satisfies the equation.
With our solution \(n = 180\), we substitute it back into the original equation \(-\frac{1}{6} \times 180 = -30\). If both sides remain equal, then the solution is correct.
Performing this check:

• Calculate \(-\frac{1}{6} \times 180\), which should give \(-30\).
• The original equation was \(-\frac{1}{6} n = -30\).
• Both sides match: the left side equals \(-30\), confirming our solution.

This step is essential for ensuring accuracy, especially in more complex equations. Being diligent in checking helps prevent errors and cements your understanding of the equation solving process.