To solve the equation [tex]\(-\frac{1}{2} w - \frac{3}{5} = \frac{1}{5} w\)[/tex], we follow these steps:
1. Isolate the variable term: Combine like terms by getting all the [tex]\(w\)[/tex] terms on one side of the equation.
[tex]\[
-\frac{1}{2} w - \frac{3}{5} = \frac{1}{5} w
\][/tex]
Add [tex]\(\frac{1}{2} w\)[/tex] to both sides to move all [tex]\(w\)[/tex] terms to one side:
[tex]\[
-\frac{1}{2} w + \frac{1}{2} w - \frac{3}{5} = \frac{1}{5} w + \frac{1}{2} w
\][/tex]
This simplifies to:
[tex]\[
-\frac{3}{5} = \frac{1}{5} w + \frac{1}{2} w
\][/tex]
2. Combine like terms on the right side: Convert [tex]\(\frac{1}{2} w\)[/tex] to a common denominator to combine it with [tex]\(\frac{1}{5} w\)[/tex].
[tex]\[
\frac{1}{2} w = \frac{5}{10} w = \frac{2.5}{5} w
\][/tex]
The right side becomes:
[tex]\[
\frac{1}{5} w + \frac{2.5}{5} w = \frac{3.5}{5} w
\][/tex]
3. Simplify the combined terms:
[tex]\[
-\frac{3}{5} = \frac{3.5}{5} w
\][/tex]
This equation balances out as:
[tex]\[
-\frac{3}{5} = \frac{7}{10} w
\][/tex]
4. Solve for [tex]\(w\)[/tex]: Isolate [tex]\(w\)[/tex] by multiplying both sides by the reciprocal of [tex]\(\frac{7}{10}\)[/tex]:
[tex]\[
w = \frac{-3/5}{7/10} = -\frac{3}{5} \times \frac{10}{7} = -\frac{3 \times 2}{5 \times 7} = -\frac{6}{35}
\][/tex]
5. Check the final solution:
The simplified result indicates:
[tex]\[
w = -0.857142857142857\ldots
\][/tex]
Thus, the solution to the equation is:
[tex]\[
w = -0.857142857142857
\][/tex]
