Problem 129 - Introductory and Intermediate Algebra for College Students 4th

Question

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$3^{x}=2 x+3$$

Step-by-Step Solution

Answer

The solution to the equation \(3^x = 2x + 3\) is the x-coordinate of the point where the graphs of the functions \(y = 3^x\) and \(y = 2x + 3\) intersect. This value is finally confirmed by directly substituting it into the initial equation.

Step 1: Graph the Functions

Start by graphing each function, \(y = 3^x\) and \(y = 2x + 3\), on the same cartesian plane using a graphic calculator or utility.

Step 2: Find the Intersection Point

After the graphs have been plotted, they will intersect at certain points. Find the x-coordinate of the point where the graphs intersect.

Step 3: Direct Substitution Verification

After obtaining the x-coordinate of the intersection point, substitute this value into both sides of the equation \(3^x = 2x + 3\). If the left side equals the right side, the solution is correct. This process is known as direct substitution verification.