Problem 76 - College Algebra

Question

In Exercises \(75-82,\) 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 to the equation \(3^{x+1} = 9\) is \(x = 1\).

Step 1: Recognize the Relationship

Notice that 9 can be written as \(3^2\). Rewrite the equation as \(3^{x+1}=3^2\).

Step 2: Graphing

Graph the two sides of the equation separately with a graphing utility. Graph \(y = 3^{x+1}\) and \(y = 3^2 = 9\). You will observe that these two graphs intersect at the point (1,9).

Step 3: Find the Intersection Point

Depending on your graphing utility, you can use the 'Intersection' function to exactly find the coordinates of the intersection point. This is where the two equations are equal. Hence, the \(x\) -coordinate of this point is the solution.

Step 4: Verification

To verify the solution, \(x=1\), substitute it into the original equation, thus: \(3^{1+1}= 3^2 = 9\), which holds true, indicating that the solution is correct.

Key Concepts

Exponential Equations

Understanding exponential equations is essential for both solving problems and grasping the growth processes they represent. An exponential equation is one in which a variable appears in the exponent and can be written in the general form of a^x = b. To solve, one common strategy involves expressing both sides as powers of the same base, which simplifies the equation to one where the exponents on corresponding sides must be equal. For example, the equation 3^{x+1} = 9 is easier to solve once recognizing that 9 is 3^2, allowing us to deduce that x+1 = 2, eventually solving for x.

Simplified Steps:

• Express both sides with the same base if possible.
• Equate the exponents and solve for the variable.

This simplification is powerful because it transforms the problem into basic algebra, making the solution more accessible.

Graphing Utilities

Graphing utilities, such as graphing calculators or software, are invaluable tools for visualizing equations and their solutions. When working with exponential equations, graphing utilities allow us to plot the function and quickly identify points of interest, such as intersections which represent solutions to the equations. Placing y=3^{x+1} and y=3^2 into a graphing utility reveals their intersection, hence the solution for x.

To effectively use graphing utilities:

• Select an appropriate scale and viewing window to accurately capture the behavior of the functions.
• Use built-in features like 'Intersection' tools to pinpoint the coordinates of intersecting points precisely.

With practice, these utilities become a quick and efficient means to corroborate solutions obtained algebraically.

Solving Algebraically

Solving algebraically entails manipulating equations using algebraic operations to find a solution. When faced with an equation like 3^{x+1} = 9, algebra offers a way to isolate the variable and solve step-by-step. By applying logarithmic operations or using properties of exponents, we can transition an exponential equation to a linear one, which is easier to manage.

Procedure for algebraic solutions:

• Rewrite the equation in a form that isolates the variable, if possible.
• Utilize logarithms if the bases cannot be made the same.
• Verify every solution in original equation due to possible extraneous solutions.

Solving algebraically also provides students with a strong foundation in the logical structure of mathematics, underscoring why a particular solution works.

Verification by Substitution

After finding a potential solution to an equation, verification by substitution ensures its validity. This process involves replacing the variable in the original equation with the solution and checking if the equation balances. For the given example, substituting x=1 back into 3^{x+1} = 9 results in 3^{1+1} = 3^2, which simplifies to 9 = 9. This confirms the solution is correct.

Why verify?

• To catch any possible mistakes made while solving.
• To ensure that the solution satisfies the original conditions of the problem.

Especially with more complex equations, this step is critical as it prevents the acceptance of incorrect or extraneous solutions.