Problem 126 - College Algebra

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 is \(x=1\).

Step 1: Rewrite the Equation

First, rewrite the right hand side of the equation in base 3. That is, rewrite 9 as \(3^2\). So, the equation \(3^{x + 1} = 9\) becomes \(3^{x + 1} = 3 ^ 2\).

Step 2: Compare the Exponents

When two exponential expressions with the same base are equal, their exponents must also be equal. Therefore, we can set the exponent of the left hand side equal to the exponent of the right hand side, that is: (x+1) = 2.

Step 3: Solve for x

To solve for x, subtract 1 from both sides of the equation (x+1)=2. This gives us x = 2 - 1, so x = 1.

Step 4: Verify

To verify, substitute x = 1 back into the original equation \(3^{x+1} = 9\). This replaces x with 1 and gives \(3^{1+1}=9\), which simplifies to \(3^2=9\), confirming our solution.

Key Concepts

Solving Exponential Equations

When it comes to solving exponential equations, students must remember that the goal is to find the value of the variable that makes the equation true. Exponential equations feature a variable in the exponent, as seen in the equation \(3^{x+1}=9\). The first step is often expressing both sides of the equation with the same base, making it easier to compare the exponents.

• Identify the base on both sides of the equation.
• Use properties of exponents to express each side with the same base if they are different.
• Set the exponents equal to each other, since bases are the same.
• Solve the resulting equation to get the value of the variable.

Applying these steps makes it possible to get to the heart of the equation—finding the power that the base needs to be raised to. In our exercise, we rewrite 9 as \(3^{2}\), allowing us to set \(x + 1 = 2\) and solve for \(x\).

Graphing Utility

A graphing utility is a powerful tool which provides a visual representation of equations. It's particularly handy when working with complex or non-linear equations, like exponential functions.

• Enter each side of the equation in the graphing utility as separate functions.
• Adjust the viewing window to ensure the graphs intersect within the visible area.
• Identify the point of intersection; the x-coordinate represents the solution.

Note that graphing can offer a visual check but may not always give an exact value due to pixel limitations, making algebraic solutions essential.

In our textbook problem, graphing \(3^{x+1}\) and \(9\) in the same window can help us to visualize the solution, but we ultimately rely on algebra for precision.

Verify Solutions

Verifying solutions is an essential and concluding part of solving equations. After obtaining a solution algebraically or graphically, one must ensure it is correct. Verification consists of substituting the solution back into the original equation to test whether it holds true.

• Substitute the value of the variable back into the original equation.
• Simplify the equation to see if both sides are equal.
• If they match, the solution is verified; if not, re-evaluate your earlier steps.

Direct verification prevents errors and confirms the accuracy of solutions. As we did in the exercise by substituting \(x = 1\) in to the original equation, we verified that \(3^{x+1} = 9\) is a true statement.

Direct Substitution

Direct substitution is a straightforward method used in the verification process and consists in replacing variables with their computed values to ensure that the initial equation holds true.

In practice:

• Pick the obtained solution for the variable.
• Replace the variable in the original equation with this solution.
• Simplify the expression to check if it equals the other side of the equation.

This method proves to be quite reliable and tangible evidence of a correct solution, when applied to our exercise, substituting \(x = 1\) directly into \(3^{x+1}\) simplifies to \(3^2\), thus making the equation true as \(3^2 = 9\). It reinforces the solution found and serves as a good habit for students to ensure their answers are accurate.