Q. 78 - Precalculus Enhanced with Graphing Utilities

Question

Solve given equation and verify your result using graphing utility.

$e^{3x} = e^{2-x}$

Step-by-Step Solution

Answer

The solution set is $\left\{\frac{1}{2}\right\}$ and it is verified through graph.

Step 1. Given information

 The given equation is :

$e^{3x} = e^{2-x}$

We have to solve for x and verify using graphing utility.

Step 2. Comparing the expressions having same bases and solving for x .

Since bases are same , therefore the exponential powers should also be the same.

$$\begin{gathered} 3x = 2-x \\ 3x + x = 2 \\ 4x = 2 \\ x=\frac{2}{4} \\ x=\frac{1}{2} \end{gathered}$$

Step 3. Verification using graphing utility

Graph 

$$\begin{gathered} Y_{1 }= e^{3x} \\ {Y}_2 = e^{2-x} \end{gathered}$$

Determine the point of intersection.

The graph intersects at $x=\frac{1}{2}$.

So the solution set is $\left\{\frac{1}{2}\right\}$.