Problem 126 - College Algebra Essentials

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. $$3^{x+1}=9$$

Step-by-Step Solution

Answer

The solution of the equation \(3^{x+1} = 9\) is x = 1.

Step 1: Graphically Solving the Exponential Equation

Use a graphing utility to plot \(y = 3^{x+1}\) and \(y = 9\). Where the two graphs intersect, that is the solution.

Step 2: Identifying the Intersection Point

The two graphs intersect at the point \((x, 9)\), where \(x\) is the solution to the equation. Make sure to verify the exact \(x\) value using your graphing tool.

Step 3: Solving Algebraically

Alternatively, the equation can also be solved algebraically. By knowing that 9 can be expressed as \(3^2\), the equation \(3^{x+1} = 3^2\) can be rewritten, by using the property that \(a^{m} = a^{n}\) implies \(m = n\). Therefore the solution to the equation is \(x+1 = 2\), and solving for \(x\) gives \(x = 1\).

Step 4: Verification through Substitution

To verify the solution, substitute \(x = 1\) into the original equation. Plugging it in gives \(3^{1+1} = 9\) which simplifies to \(3^2 = 9\), confirming that the solution is correct.

Key Concepts

Graphing Utility

When solving exponential equations, a graphing utility can be incredibly useful. These tools allow you to visualize equations by creating precise graphs of functions. For the equation \(3^{x+1} = 9\), using a graphing utility means plotting two different equations simultaneously:

• \(y = 3^{x+1}\)
• \(y = 9\)

Plotting these on the same graph creates a visual representation where you can see their behavior across different \(x\)-values.
This can be particularly handy as it quickly highlights areas, such as intersection points, where both equations hold the same value. This saves time compared to manually calculating where these functions might equal each other.

Intersection Point

The intersection point of two graphs represents the x-value where both equations are equal. In our example of the exponential equation, we look for the place where both functions, \(y = 3^{x+1}\) and \(y = 9\), intersect. This point is crucial as it directly provides the solution to the equation.
By examining the graph provided by your graphing utility, the intersection point is the point where the curves cross each other. For this particular example, this occurs when \(x = 1\), making it the solution for the equation \(3^{x+1}=9\).
Ensure you use the precise functionalities of your graphing tool to find the exact x-coordinate, as this confirms the solution found visually matches algebraically.

Substitution

Once an intersection point has been identified, substitution is a valuable way to verify the solution. Substitution involves plugging the solution back into the original equation to ensure that both sides equal each other.
In our example, with \(x = 1\) identified from the intersection point, substitute back into the original equation:

• Original equation: \(3^{x+1} = 9\)
• Substitute \(x = 1\): \(3^{1+1} = 9\)
• Calculates to: \(3^2 = 9\)

This confirms that both sides of the equation are equal and verifies that \(x = 1\) is indeed a correct solution.

Algebraic Solving

While graphing is a fantastic visual tool, solving algebraically can reinforce your understanding and provide a more versatile skill set for different types of problems. In this case, algebraic solving involves recognizing that the number \(9\) can be expressed as a power of 3, specifically \(3^2\).
This allows the equation \(3^{x+1} = 9\) to be rewritten as \(3^{x+1} = 3^2\). Here, a key property of exponents comes into play: if the bases are the same, the exponents must be equal for the equation to hold. Therefore, you set up the equation:

• \(x+1 = 2\)
• Solve for \(x\): \(x = 2 - 1 = 1\)

Using algebra, we reach the same conclusion \(x = 1\) as the solution, consistent with the graphing result. This approach is analytical and helps strengthen mathematical reasoning.