Solve: $\frac{1}{3 a^2}-\frac{1}{a}=\frac{1}{6 a^2}$
Answer (2)
Multiply both sides of the equation by 6 a 2 to eliminate fractions.
Simplify the equation to 2 − 6 a = 1 .
Isolate the term with a to get − 6 a = − 1 .
Solve for a by dividing both sides by -6, resulting in a = 6 1 .
Explanation
Problem Setup We are given the equation 3 a 2 1 − a 1 = 6 a 2 1 and we want to solve for a .
Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by 6 a 2 :
6 a 2 ( 3 a 2 1 − a 1 ) = 6 a 2 ( 6 a 2 1 ) Distributing on the left side gives: 6 a 2 ⋅ 3 a 2 1 − 6 a 2 ⋅ a 1 = 1 Simplifying, we get: 2 − 6 a = 1
Isolating the Term with a Subtracting 2 from both sides, we have: − 6 a = 1 − 2 − 6 a = − 1
Solving for a Dividing both sides by -6, we solve for a :
a = − 6 − 1 = 6 1
Final Answer Thus, the solution is a = 6 1 .
Examples
In electrical engineering, this type of equation can appear when analyzing circuits with resistors and capacitors. For example, the variable a could represent a time constant in an RC circuit, and solving for a helps determine the circuit's response time. Understanding how to manipulate and solve such equations is crucial for designing and troubleshooting electronic systems.
To solve the equation, we multiply both sides by 6 a 2 to eliminate fractions, leading to a simpler equation. After isolating a , we find that a = 6 1 .
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