Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![**Title: Solving an Exponential Equation and Verifying Graphically**
**Objective:** Solve the equation algebraically and verify the solution by using graphical methods.
**Equation:**
\[
3e^{7x} = 651
\]
**Steps to Solve Algebraically:**
1. **Isolate the Exponential Term**: Divide both sides by 3.
\[
e^{7x} = \frac{651}{3}
\]
2. **Simplify the Equation**:
\[
e^{7x} = 217
\]
3. **Apply the Natural Logarithm**: Take the natural logarithm of both sides to solve for \(x\).
\[
\ln(e^{7x}) = \ln(217)
\]
4. **Use Logarithm Properties**: Simplify using \(\ln(e^a) = a\).
\[
7x = \ln(217)
\]
5. **Solve for \(x\)**:
\[
x = \frac{\ln(217)}{7}
\]
**Final Solution:**
\[
x \approx \boxed{\text{(Insert approximate value here, rounded to the nearest thousandth as needed)}}
\]
**Graphical Verification:**
To verify the solution, graph the functions \(y = 3e^{7x}\) and \(y = 651\) on the same coordinate plane. The x-coordinate of the intersection point will confirm the algebraic solution.
**Rounding Instructions:**
Round your final answer for \(x\) to the nearest thousandth to ensure precision and consistency.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70a8cbee-10b6-427b-ac1c-41a66a493bbd%2F9b53dda8-2c7d-4415-899e-b10d192b13f9%2Fyw2uka5_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Solving an Exponential Equation and Verifying Graphically**
**Objective:** Solve the equation algebraically and verify the solution by using graphical methods.
**Equation:**
\[
3e^{7x} = 651
\]
**Steps to Solve Algebraically:**
1. **Isolate the Exponential Term**: Divide both sides by 3.
\[
e^{7x} = \frac{651}{3}
\]
2. **Simplify the Equation**:
\[
e^{7x} = 217
\]
3. **Apply the Natural Logarithm**: Take the natural logarithm of both sides to solve for \(x\).
\[
\ln(e^{7x}) = \ln(217)
\]
4. **Use Logarithm Properties**: Simplify using \(\ln(e^a) = a\).
\[
7x = \ln(217)
\]
5. **Solve for \(x\)**:
\[
x = \frac{\ln(217)}{7}
\]
**Final Solution:**
\[
x \approx \boxed{\text{(Insert approximate value here, rounded to the nearest thousandth as needed)}}
\]
**Graphical Verification:**
To verify the solution, graph the functions \(y = 3e^{7x}\) and \(y = 651\) on the same coordinate plane. The x-coordinate of the intersection point will confirm the algebraic solution.
**Rounding Instructions:**
Round your final answer for \(x\) to the nearest thousandth to ensure precision and consistency.
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