Sure, let's work through the problem step by step.
We are given the function:
[tex]\[ f(x) = -\frac{1}{3}x + 7 \][/tex]
We want to determine the value of [tex]\( x \)[/tex] that would result in an output of [tex]\( \frac{2}{3} \)[/tex]. This means we need to solve for [tex]\( x \)[/tex] when:
[tex]\[ f(x) = \frac{2}{3} \][/tex]
Start with the equation:
[tex]\[ \frac{2}{3} = -\frac{1}{3} x + 7 \][/tex]
First, isolate the term with [tex]\( x \)[/tex]. To do this, subtract 7 from both sides of the equation:
[tex]\[ \frac{2}{3} - 7 = -\frac{1}{3} x \][/tex]
Convert 7 to a fraction with the same denominator as [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ 7 = \frac{21}{3} \][/tex]
Now the equation becomes:
[tex]\[ \frac{2}{3} - \frac{21}{3} = -\frac{1}{3} x \][/tex]
Combine the fractions on the left-hand side:
[tex]\[ \frac{2 - 21}{3} = -\frac{1}{3} x \][/tex]
Simplify the numerator:
[tex]\[ \frac{-19}{3} = -\frac{1}{3} x \][/tex]
To solve for [tex]\( x \)[/tex], get rid of the fraction by multiplying both sides of the equation by -3:
[tex]\[ -3 \left( \frac{-19}{3} \right) = (-3) \left( -\frac{1}{3} x \right) \][/tex]
This simplifies to:
[tex]\[ 19 = x \][/tex]
Therefore, the input that would give an output value of [tex]\( \frac{2}{3} \)[/tex] is:
[tex]\[ x = 19 \][/tex]
