91Ó°ÊÓ

When solving linear equations, one crucial step is to isolate the variable, which means getting the variable by itself on one side of the equation. In the given equation \(\frac{1}{5}p = -3\), the goal is to solve for \(p\). To isolate \(p\), you need to eliminate any coefficients or constants attached to it. Here, the coefficient is \(\frac{1}{5}\).

We can do this by multiplying both sides of the equation by the reciprocal of \(\frac{1}{5}\), which is 5. This step ensures that \(p\) is left alone:

\[ 5 \cdot \frac{1}{5} p = 5 \cdot (-3) = -15 \]

Thus, \(p = -15\). By isolating the variable, we've solved the equation for \(p\).

A reciprocal is a number which, when multiplied with another number, results in 1. For example, the reciprocal of \(\frac{1}{5}\) is 5 because \(\frac{1}{5} \cdot 5 = 1\). In solving equations, the reciprocal helps to balance the equation, allowing us to isolate the variable.

In the case of \(\frac{1}{5}p = -3\), we used 5, the reciprocal of \(\frac{1}{5}\), to get rid of the fraction and isolate \(p\). Multiplying both sides by 5, the equation simplifies as follows:

\[ 5 \cdot \frac{1}{5} p = 5 \cdot -3 \] which simplifies to \[ p = -15 \].

Knowing how to use the reciprocal is essential for solving equations involving fractions, making the process straightforward and manageable.

After solving an equation, it is vital to check the solution to ensure it is correct. This involves substituting the found value back into the original equation to see if both sides are equal.

For the equation \(\frac{1}{5} p = -3\) and solution \(p = -15\), we can verify the solution by plugging \(p = -15\) back into the original equation:

\[ \frac{1}{5} (-15) = -3 \]

Simplifying the left side, we get: \[ -3 = -3 \]. Since both sides of the equation are equal, this confirms that \(p = -15\) is indeed the correct solution.

Checking solutions ensures that no mistakes were made during the calculation, providing confidence in your answer.