Problem 130 - 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. $$5^{x}=3 x+4$$

Step-by-Step Solution

Answer

The solution of the equation \(5^{x}=3 x+4\) is the x-coordinate of the intersection point of the graphs of the functions \(y = 5^{x}\) and \(y = 3x + 4\). This value can be verified by direct substitution into the original equation.

Step 1: Graph the equation

Start by graphing the functions \(y = 5^{x}\) and \(y = 3x + 4\) separately. The intersection of these two graphs corresponds to the solution of the equation.

Step 2: Find the intersection point

Using your graphing utility, identify the point at which the two graphs intersect. The x-coordinate of this point is the solution for the equation \(5^x = 3x +4\). Let's assume this value is \(a\), where \(a\) could potentially be any real number.

Step 3: Verify solution

After obtaining the value of \(a\), substitute it into the original equation to verify if both sides match. Put \(a\) in place of \(x\) in both \(5^x\) and \(3x + 4\), and check whether \(5^a = 3a + 4\) holds true.