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Given that f(x)=k+x/(3x-2) and that f(-1/3)=1/6, find the value of k.

September 29, 2026Mathematics

ScanSolve AI Answer

Answer

-\frac{1}{6}

4 steps

Substitute $x = -\frac{1}{3}$ into the given function $f(x) = \frac{k+x}{3x-2}$ .
$f\left(-\frac{1}{3}\right) = \frac{k + \left(-\frac{1}{3}\right)}{3\left(-\frac{1}{3}\right) - 2}$
Simplify the expression for $f\left(-\frac{1}{3}\right)$ .
$f\left(-\frac{1}{3}\right) = \frac{k - \frac{1}{3}}{-1 - 2}$ $f\left(-\frac{1}{3}\right) = \frac{k - \frac{1}{3}}{-3}$
Set the simplified expression equal to $\frac{1}{6}$ , as given, and solve for $k$ .
$\frac{k - \frac{1}{3}}{-3} = \frac{1}{6}$
Multiply both sides by $-3$ :
$k - \frac{1}{3} = \frac{1}{6} \times (-3)$ $k - \frac{1}{3} = -\frac{3}{6}$ $k - \frac{1}{3} = -\frac{1}{2}$
Isolate $k$ by adding $\frac{1}{3}$ to both sides.
$k = -\frac{1}{2} + \frac{1}{3}$
Find a common denominator, which is 6:
$k = -\frac{3}{6} + \frac{2}{6}$ $k = \frac{-3+2}{6}$ $k = -\frac{1}{6}$
The value of $k$ is $\boxed{-\frac{1}{6}}$ .

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