Problem 139 - Algebra and Trigonometry

Question

Without actually solving the equation, describe how to solve $$ 3 \tan x-2=5 \tan x-1 $$

Step-by-Step Solution

Answer

To solve the equation \(3 \tan x - 2 = 5 \tan x - 1\), one would first rearrange the equation to result in \(2 \tan x = 1\), then solve for \(\tan \), resulting in \(x = \tan^{-1}(\frac{1}{2})\). The actual solution is not required in this instance.

Step 1: Understand the Equation

The given equation is a trigonometric equation because it involves the tangent function. It's structured like a simple linear equation, but involves trigonometric functions.

Step 2: Rearrange the Equation

Start by bringing like terms together to one side of the equation. This means subtracting \(3 \tan x\) from both sides of the equation: \[5 \tan x - 3 \tan x = 2 - 1\]. The equation simplifies to \[2 \tan x = 1\].

Step 3: Isolate the variable

Now, isolate \(x\) by dividing each side by \(2\). This results in a new equation: \[\tan x = \frac{1}{2}\].

Step 4: Write the final step without solving

The final step to actually solving the equation (which is not required here) would be to find the inverse tangent of \(1/2\) - this gives the value of \(x\). Write it as: \[x = \tan^{-1}(\frac{1}{2})\].