Determine Whether the Given Values Are Solutions of the Equation
Question:
Determine whether the given values are solution of the given equation or not:
\[ x+\frac{1}{x}=\frac{13}{6} \]
(i) \(x=\frac{5}{6}\)
(ii) \(x=\frac{4}{3}\)
Solution
A value of \(x\) is a solution if, after substitution, the left-hand side becomes equal to the right-hand side.
Checking \(x=\frac{5}{6}\)
Substitute \(x=\frac{5}{6}\):
\[ \frac{5}{6}+\frac{1}{\frac{5}{6}} \]
\[ =\frac{5}{6}+\frac{6}{5} \]
\[ =\frac{25+36}{30} \]
\[ =\frac{61}{30} \]
Since \[ \frac{61}{30}\ne\frac{13}{6}, \] \(x=\frac{5}{6}\) is not a solution.
Checking \(x=\frac{4}{3}\)
Substitute \(x=\frac{4}{3}\):
\[ \frac{4}{3}+\frac{1}{\frac{4}{3}} \]
\[ =\frac{4}{3}+\frac{3}{4} \]
\[ =\frac{16+9}{12} \]
\[ =\frac{25}{12} \]
Since \[ \frac{25}{12}\ne\frac{13}{6}, \] \(x=\frac{4}{3}\) is not a solution.
Answer
Neither of the given values satisfies the equation.
\[ \boxed{x=\frac{5}{6}\text{ is not a solution and }x=\frac{4}{3}\text{ is not a solution}} \]