To isolate the variable term on one side and the constant term on the other for the given equation \( \frac{1}{5} x + \frac{1}{3} = -\frac{1}{2} x + \frac{8}{3} \), we can manipulate the equation step by step.
-
Combine like terms: Start by getting all terms with \( x \) on one side and constants on the other.
First, let's move \( -\frac{1}{2} x \) to the left side: \[ \frac{1}{5} x + \frac{1}{2} x + \frac{1}{3} = \frac{8}{3} \] This gives us: \[ \left( \frac{1}{5} + \frac{1}{2} \right)x + \frac{1}{3} = \frac{8}{3} \]
Next, let's also move \( \frac{1}{3} \) to the right side: \[ \left( \frac{1}{5} + \frac{1}{2} \right)x = \frac{8}{3} - \frac{1}{3} \]
Simplifying that: \[ \left( \frac{1}{5} + \frac{1}{2} \right)x = \frac{7}{3} \]
-
Combine \( x \) coefficients: The least common multiple of 5 and 2 is 10. \[ \frac{1}{5} = \frac{2}{10}, \quad \frac{1}{2} = \frac{5}{10} \] Therefore, \[ \frac{2}{10} + \frac{5}{10} = \frac{7}{10} \] Now, substituting back: \[ \frac{7}{10}x = \frac{7}{3} \]
-
Isolating \( x \): Dividing both sides by \( \frac{7}{10} \): \[ x = \frac{7}{3} \div \frac{7}{10} = \frac{7}{3} \cdot \frac{10}{7} = \frac{10}{3} \]
So, we can check the forms:
-
One-fifth \( x \) minus \( \frac{7}{3} \) = negative one-half: This rearranges correctly and is valid.
-
\( \frac{7}{10} x = \frac{7}{3} \): This resembles the correct formulation we arrived at.
-
Negative \( \frac{7}{3} = -\frac{7}{10} x \): This also is valid as a manipulation of the equation we derived.
-
\( x = -\frac{5}{2} x + \frac{35}{3} \): This does not match any transformations we applied readily to the original equation.
Given these evaluations, the correct equations to isolate would be:
- One-fifth \( x \) minus \( \frac{7}{3} \) = negative one-half
- \( \frac{7}{10} x = \frac{7}{3} \)
- Negative \( \frac{7}{3} = -\frac{7}{10} x \)
Thus, the valid options are first three statements:
- One-fifth \( x \) minus \( \frac{7}{3} = -\frac{1}{2} \)
- \( \frac{7}{10} x = \frac{7}{3} \)
- Negative \( \frac{7}{3} = -\frac{7}{10} x \)