Sure, let’s tackle this equation step by step:
Given equation:
[tex]\[
-\frac{1}{3}\left(\frac{2}{3} x + 5\right) = -\frac{4}{3}\left(\frac{1}{2} x - 8\right)
\][/tex]
1. Use the distributive property to expand both sides:
For the left side:
[tex]\[
-\frac{1}{3} \left(\frac{2}{3} x + 5 \right)
\][/tex]
This distributes to:
[tex]\[
-\frac{1}{3} \cdot \frac{2}{3} x - \frac{1}{3} \cdot 5
\][/tex]
Which simplifies to:
[tex]\[
-\frac{2}{9} x - \frac{5}{3}
\][/tex]
For the right side:
[tex]\[
-\frac{4}{3}\left(\frac{1}{2} x - 8 \right)
\][/tex]
This distributes to:
[tex]\[
-\frac{4}{3} \cdot \frac{1}{2} x + -\frac{4}{3} \cdot -8
\][/tex]
Which simplifies to:
[tex]\[
-\frac{4}{6} x + \frac{32}{3}
\][/tex]
And further simplifying:
[tex]\[
-\frac{2}{3} x + \frac{32}{3}
\][/tex]
Hence, the equation now is:
[tex]\[
-\frac{2}{9} x - \frac{5}{3} = -\frac{2}{3} x + \frac{32}{3}
\][/tex]
2. Isolate the variable [tex]\( x \)[/tex]:
Let's add [tex]\(\frac{2}{3} x\)[/tex] to both sides:
[tex]\[
-\frac{2}{9} x + \frac{2}{3} x - \frac{5}{3} = \frac{32}{3}
\][/tex]
Combine the [tex]\( x \)[/tex]-terms (note: converting [tex]\(\frac{2}{3}\)[/tex] to a common denominator with [tex]\(\frac{2}{9}\)[/tex]):
[tex]\[
\left(-\frac{2}{9} + \frac{6}{9}\right)x - \frac{5}{3} = \frac{32}{3}
\][/tex]
Simplifying the coefficients of [tex]\( x \)[/tex]:
[tex]\[
\frac{4}{9} x - \frac{5}{3} = \frac{32}{3}
\][/tex]
3. Isolate [tex]\( x \)[/tex] further:
Add [tex]\(\frac{5}{3}\)[/tex] to both sides:
[tex]\[
\frac{4}{9} x = \frac{32}{3} + \frac{5}{3}
\][/tex]
Combine the fractions on the right side:
[tex]\[
\frac{4}{9} x = \frac{37}{3}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by [tex]\(\frac{4}{9}\)[/tex]:
[tex]\[
x = \frac{\frac{37}{3}}{\frac{4}{9}} = \frac{37}{3} \times \frac{9}{4} = \frac{37 \cdot 9}{3 \cdot 4} = \frac{37 \cdot 3}{4} = \frac{111}{4}
\][/tex]
5. Review the intermediary equations given:
[tex]\[
-\frac{2}{9} x - \frac{5}{3} = -\frac{2}{3} x + \frac{32}{3} \, \quad \text{(first step)}
\][/tex]
Next, combining like terms:
[tex]\[
\frac{4}{9} x = \frac{37}{3} \, \quad \text{(isolating \( x \))}
\][/tex]
Which upon verification shows:
[tex]\[
-\frac{2}{9} x + \frac{2}{3} x - \frac{37}{3} = 0 \, \quad \text{(simplified step)}
\][/tex]
Further isolation:
[tex]\[
-\frac{4}{9} x = \frac{27}{3} \, \quad \text{(incorrect as context)}
\][/tex]
Based on the detailed solution and the numerical verification, we conclude the equations that correctly use the distributive property and the properties of equality to isolate the variable term:
\- [tex]\(-\frac{2}{9} x - \frac{5}{3} = -\frac{2}{3} x + \frac{32}{3}\)[/tex]
\- [tex]\(\frac{4}{9} x = \frac{37}{3}\)[/tex]
\- [tex]\(-\frac{2}{9} x + \frac{2}{3} x - \frac{37}{3} = 0\)[/tex]
