To determine the input [tex]\(x\)[/tex] that would give an output of [tex]\(\frac{2}{3}\)[/tex] for the function [tex]\(f(x) = -\frac{1}{3} x + 7\)[/tex], follow these steps:
1. Set Up the Equation:
Given [tex]\(f(x) = -\frac{1}{3} x + 7\)[/tex] and we want [tex]\(f(x) = \frac{2}{3}\)[/tex],
[tex]\[
\frac{2}{3} = -\frac{1}{3} x + 7
\][/tex]
2. Isolate the Linear Term:
Subtract 7 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
\frac{2}{3} - 7 = -\frac{1}{3} x
\][/tex]
3. Simplify:
Combine the constants on the left side:
[tex]\[
\frac{2}{3} - \frac{21}{3} = -\frac{1}{3} x
\][/tex]
Simplifying the left-hand side:
[tex]\[
\frac{2 - 21}{3} = -\frac{1}{3} x
\][/tex]
[tex]\[
-\frac{19}{3} = -\frac{1}{3} x
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Multiply both sides by -3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -3 \left( -\frac{19}{3} \right)
\][/tex]
[tex]\[
x = 19
\][/tex]
Thus, the input [tex]\(x\)[/tex] that results in the output [tex]\( \frac{2}{3} \)[/tex] is [tex]\( \boxed{19} \)[/tex].
