Solve [tex]$\frac{1}{x^2}=\frac{1}{6 x^2}+\frac{1}{6 x}$[/tex].
Answer (2)
Multiply both sides of the equation by 6 x 2 to eliminate the fractions.
Simplify the equation to 6 = 1 + x .
Solve for x by subtracting 1 from both sides.
The solution is 5 .
Explanation
Problem Analysis We are given the equation x 2 1 = 6 x 2 1 + 6 x 1 and asked to solve for x .
Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by 6 x 2 . This gives us: 6 x 2 ⋅ x 2 1 = 6 x 2 ⋅ ( 6 x 2 1 + 6 x 1 ) 6 = 1 + x
Solving for x Subtracting 1 from both sides, we get: x = 6 − 1 x = 5
Final Answer Therefore, the solution is x = 5 .
Examples
Understanding how to solve rational equations is crucial in many fields, such as physics and engineering. For example, when analyzing electrical circuits, you might encounter equations involving reciprocals of variables. Solving these equations allows you to determine unknown resistances or currents in the circuit. Similarly, in physics, you might use rational equations to model the motion of objects or the behavior of waves. Mastering these algebraic techniques provides a foundation for tackling more complex problems in these areas.
To solve the equation x 2 1 = 6 x 2 1 + 6 x 1 , we first eliminate the fractions by multiplying through by 6 x 2 , simplifying to 6 = 1 + x . Solving for x gives us the solution x = 5 .
;
