What is the solution to the linear equation?
$\frac{2}{5}+p=\frac{4}{5}+\frac{3}{5} p$
The problem asks us to solve the linear equation \(\frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p\) for the variable \(p\). We have a single unknown \(p\) in a linear equation, which can be rearranged by isolating the terms involving \(p\) on one side and the constant values on the other. This will allow us to simplify the equation and determine the value of \(p\).
To isolate the terms with \(p\), we subtract \(\frac{3}{5}p\) from both sides of the equation:
\[\left(p - \frac{3}{5}p\right) + \frac{2}{5} = \frac{4}{5}\]
Since \(p\) is the same as \(\frac{5}{5}p\), we have \(\frac{5}{5}p - \frac{3}{5}p = \frac{2}{5}p\). So, the equation becomes:
\[\frac{2}{5}p + \frac{2}{5} = \frac{4}{5}\]
Next, we isolate the term with \(p\) by subtracting \(\frac{2}{5}\) from both sides of the equation:
\[\frac{2}{5}p = \frac{4}{5} - \frac{2}{5}\]
\[\frac{2}{5}p = \frac{2}{5}\]
To solve for \(p\), we divide both sides by the coefficient \(\frac{2}{5}\), which is equivalent to multiplying both sides by \(\frac{5}{2}\):
\[p = \frac{2}{5} \times \frac{5}{2}\]
\[p = 1\]
Thus, the solution to the equation is \(p = 1\). Comparing this result to the given options, we find that \(p=1\) is the correct choice.
Linear equations like these are essential in real-world scenarios such as calculating hourly wages or mixing ingredients for a recipe. For instance, if you are determining how many hours of work you need to finish a project where you are already paid a base fee and an additional hourly rate, you are setting up a linear equation. Solving for the unknown allows you to plan your time and budget effectively, ensuring that both sides of the "equation"—income and expenses—balance out correctly.