Finite Mathematics & Its Applications (12th Edition)
12th Edition
ISBN: 9780134507125
Author: Goldstein
Publisher: PEARSON
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Textbook Question
Chapter 1, Problem 2FCCE
What is meant by the graph of an equation in x and y?
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Total marks 16
5.
Let (,,P) be a probability space and let X : → R be a random
variable whose probability density function is given by f(x) = }}|x|e¯|×| for
x Є R.
(i)
(ii)
Find the characteristic function of the random variable X.
[8 Marks]
Using the result of (i), calculate the first two moments of the
random variable X, i.e., E(X") for n = 1, 2.
(iii) What is the variance of X?
[6 Marks]
[2 Marks]
Refer to page 12 for a problem on solving a homogeneous differential equation.
Instructions:
• Simplify the equation into a homogeneous form.
Use appropriate substitutions to reduce complexity.
Solve systematically and verify the final result with clear back-substitutions.
Link:
[https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]
Refer to page 36 for solving a bang-bang control problem.
Instructions:
•
Formulate the problem, identifying the control constraints.
•
•
Apply Pontryagin's Maximum Principle to derive the switching conditions.
Clearly illustrate the switching points in the control trajectory. Verify the solution satisfies the
optimality criteria.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]
Chapter 1 Solutions
Finite Mathematics & Its Applications (12th Edition)
Ch. 1.1 - Plot the point (500,200).Ch. 1.1 - 2. Is the point on the graph of the linear...Ch. 1.1 - In Exercises 1–8, plot the given point.
1.
Ch. 1.1 - In Exercises 18, plot the given point. (1,4)Ch. 1.1 - In Exercises 1–8, plot the given point.
3.
Ch. 1.1 - In Exercises 18, plot the given point. (2,0)Ch. 1.1 - In Exercises 18, plot the given point. (2,1)Ch. 1.1 - In Exercises 18, plot the given point. (1,52)Ch. 1.1 - In Exercises 1–8, plot the given point.
7.
Ch. 1.1 - In Exercises 1–8, plot the given point.
8.
Ch. 1.1 - What are the coordinates of the point Q in Fig. 7?Ch. 1.1 - What are the coordinates of the point P in Fig. 7?Ch. 1.1 - In Exercises 11–14, determine whether the point is...Ch. 1.1 - In Exercises 1114, determine whether the point is...Ch. 1.1 - In Exercises 11–14, determine whether the point is...Ch. 1.1 - In Exercises 1114, determine whether the point is...Ch. 1.1 - In Exercises 15–18, each linear equation is in the...Ch. 1.1 - In Exercises 1518, each linear equation is in the...Ch. 1.1 - In Exercises 1518, each linear equation is in the...Ch. 1.1 - In Exercises 1518, each linear equation is in the...Ch. 1.1 - In Exercises 19–22, write each linear equation in...Ch. 1.1 - In Exercises 1922, write each linear equation in...Ch. 1.1 - In Exercises 1922, write each linear equation in...Ch. 1.1 - In Exercises 19–22, write each linear equation in...Ch. 1.1 - In Exercises 2326, find the x-intercept and the...Ch. 1.1 - In Exercises 2326, find the x-intercept and the...Ch. 1.1 - In Exercises 2326, find the x-intercept and the...Ch. 1.1 - In Exercises 2326, find the x-intercept and the...Ch. 1.1 - In Exercises 2734, graph the given linear...Ch. 1.1 - In Exercises 2734, graph the given linear...Ch. 1.1 - In Exercises 27–34, graph the given linear...Ch. 1.1 - In Exercises 2734, graph the given linear...Ch. 1.1 - In Exercises 27–34, graph the given linear...Ch. 1.1 - In Exercises 2734, graph the given linear...Ch. 1.1 - In Exercises 27–34, graph the given linear...Ch. 1.1 - In Exercises 2734, graph the given linear...Ch. 1.1 - Which of the following equations describe the same...Ch. 1.1 - Which of the following equations describe the same...Ch. 1.1 - 37. Each of the lines in Fig. 8 is the graph of...Ch. 1.1 - 38. Which of the following equations is graphed in...Ch. 1.1 - 39. Heating Water The temperature of water in a...Ch. 1.1 - Life Expectancy The average life expectancy y of a...Ch. 1.1 - 41. Cigarette Consumption The worldwide...Ch. 1.1 - 42. Ecotourism Income In a certain developing...Ch. 1.1 - Insurance Rates Yearly car insurance rates have...Ch. 1.1 - Simple Interest If $1000 is deposited at 3% simple...Ch. 1.1 - 45. College Freshmen The percentage, y, of college...Ch. 1.1 - College Freshmen The percentage, y, of college...Ch. 1.1 - College Tuition Average tuition (including room...Ch. 1.1 - 48. Bachelor’s Degrees The number of bachelor’s...Ch. 1.1 - 49. Find an equation of the line having...Ch. 1.1 - Find an equation of the line having x-intercept...Ch. 1.1 - Find an equation of the line having y-intercept...Ch. 1.1 - Find an equation of the line having x-intercept...Ch. 1.1 - 53. What is the equation of the x-axis?
Ch. 1.1 - 54. Can a line other than the x-axis have more...Ch. 1.1 - 55. What is the general form of the equation of a...Ch. 1.1 - What is the general form of the equation of a line...Ch. 1.1 - In Exercises 5760, find a general form of the...Ch. 1.1 - In Exercises 5760, find a general form of the...Ch. 1.1 - In Exercises 5760, find a general form of the...Ch. 1.1 - In Exercises 57–60, find a general form of the...Ch. 1.1 - Show that the straight line with x-intercept (a,0)...Ch. 1.1 - Use the result of Exercise 61 to find a general...Ch. 1.1 - In Exercises 6370, give the equation of a line...Ch. 1.1 - In Exercises 6370, give the equation of a line...Ch. 1.1 - In Exercises 63–70, give the equation of a line...Ch. 1.1 - In Exercises 63–70, give the equation of a line...Ch. 1.1 - In Exercises 63–70, give the equation of a line...Ch. 1.1 - In Exercises 6370, give the equation of a line...Ch. 1.1 - In Exercises 63–70, give the equation of a line...Ch. 1.1 - In Exercises 6370, give the equation of a line...Ch. 1.1 - 71. The lines with equations and have the same...Ch. 1.1 - The lines with equations 6x3y=9 and y=4x+b have...Ch. 1.1 - In Exercises 7376, (a) graph the line, (b) use the...Ch. 1.1 - In Exercises 73–76, (a) graph the line, (b) use...Ch. 1.1 - In Exercises 7376, (a) graph the line, (b) use the...Ch. 1.1 - In Exercises 73–76, (a) graph the line, (b) use...Ch. 1.1 - In Exercises 77 and 78, determine an appropriate...Ch. 1.1 - In Exercises 77 and 78, determine an appropriate...Ch. 1.2 - Suppose that the revenue y from selling x units of...Ch. 1.2 - Suppose that the revenue y from selling x units of...Ch. 1.2 - In Exercises 1–6, find the slope of the line...Ch. 1.2 - In Exercises 1–6, find the slope of the line...Ch. 1.2 - In Exercises 1–6, find the slope of the line...Ch. 1.2 - In Exercises 16, find the slope of the line having...Ch. 1.2 - In Exercises 16, find the slope of the line having...Ch. 1.2 - In Exercises 1–6, find the slope of the line...Ch. 1.2 - In Exercises 710, plot each pair of points, draw...Ch. 1.2 - In Exercises 710, plot each pair of points, draw...Ch. 1.2 - In Exercises 7–10, plot each pair of points, draw...Ch. 1.2 - In Exercises 710, plot each pair of points, draw...Ch. 1.2 - What is the slope of any line parallel to the...Ch. 1.2 - Why doesnt it make sense to talk about the slope...Ch. 1.2 - In Exercises 1316, graph the given linear equation...Ch. 1.2 - In Exercises 13–16, graph the given linear...Ch. 1.2 - In Exercises 1316, graph the given linear equation...Ch. 1.2 - In Exercises 1316, graph the given linear equation...Ch. 1.2 - In Exercises 17–24, find the equation of line...Ch. 1.2 - In Exercises 17–24, find the equation of line...Ch. 1.2 - In Exercises 17–24, find the equation of line...Ch. 1.2 - In Exercises 1724, find the equation of line LCh. 1.2 - In Exercises 17–24, find the equation of line...Ch. 1.2 - In Exercises 17–24, find the equation of line...Ch. 1.2 - Prob. 23ECh. 1.2 - Prob. 24ECh. 1.2 - In Exercises 25–28, give the slope–intercept form...Ch. 1.2 - Prob. 26ECh. 1.2 - In Exercises 2528, give the slopeintercept form of...Ch. 1.2 - In Exercises 2528, give the slopeintercept form of...Ch. 1.2 - 29. Find the equation of the line passing through...Ch. 1.2 - 30. Find the equation of the line passing through...Ch. 1.2 - 31. Find the y-intercept of the line passing...Ch. 1.2 - 32. Find the y-intercept of the line passing...Ch. 1.2 - 33. Find the equation of the line passing through ...Ch. 1.2 - Find the equation of the line passing through the...Ch. 1.2 -
35. Cost Curve A manufacturer has fixed costs...Ch. 1.2 - Demand Curve The price p that must be set in order...Ch. 1.2 - 37. Boiling Point of Water At sea level, water...Ch. 1.2 - Cricket Chirps Biologists have found that the...Ch. 1.2 - Prob. 39ECh. 1.2 - Exercises 4042 are related. Cost Equation Suppose...Ch. 1.2 - Exercises 40–42 are related.
41. Revenue Equation...Ch. 1.2 - Exercises 4042 are related. Profit Equation...Ch. 1.2 - Prob. 43ECh. 1.2 - Cash Reserves A corporation receives payment for a...Ch. 1.2 - 45. Weekly Pay A furniture salesperson earns $220...Ch. 1.2 - 46. Weekly Pay A salesperson’s weekly pay depends...Ch. 1.2 - In Exercises 47–58, find an equation for each of...Ch. 1.2 - In Exercises 47–58, find an equation for each of...Ch. 1.2 - In Exercises 4758, find an equation for each of...Ch. 1.2 - In Exercises 4758, find an equation for each of...Ch. 1.2 - In Exercises 47–58, find an equation for each of...Ch. 1.2 - In Exercises 4758, find an equation for each of...Ch. 1.2 - In Exercises 47–58, find an equation for each of...Ch. 1.2 - In Exercises 4758, find an equation for each of...Ch. 1.2 - In Exercises 4758, find an equation for each of...Ch. 1.2 - In Exercises 47–58, find an equation for each of...Ch. 1.2 - Prob. 57ECh. 1.2 - Prob. 58ECh. 1.2 - Prob. 59ECh. 1.2 - In each of Exercises 5962, we specify a line by...Ch. 1.2 - Prob. 61ECh. 1.2 - Prob. 62ECh. 1.2 - Prob. 63ECh. 1.2 - Prob. 64ECh. 1.2 - Prob. 65ECh. 1.2 - Prob. 66ECh. 1.2 - Prob. 67ECh. 1.2 - Prob. 68ECh. 1.2 - Prob. 69ECh. 1.2 - Prob. 70ECh. 1.2 - Prob. 71ECh. 1.2 - 72. Dating of Artifacts An archaeologist dates a...Ch. 1.2 - Prob. 73ECh. 1.2 - 74. College Enrollments Two-year college...Ch. 1.2 - Gas Mileage A certain car gets 25 miles per gallon...Ch. 1.2 - 76. Home Health Aid Jobs According to the U.S....Ch. 1.2 - Prob. 77ECh. 1.2 - Pizza Stores According to Pizza Marketing...Ch. 1.2 - 79. Super Bowl Commercials The average cost of a...Ch. 1.2 - 80. Straight-Line Depreciation A multi-function...Ch. 1.2 - Supply Curve Suppose that 5 million tons of apples...Ch. 1.2 - Demand Curve Suppose that 5 million tons of apples...Ch. 1.2 - Show that the points (1,3),(2,4),and(3,1) are not...Ch. 1.2 - For what value of k will the three points...Ch. 1.2 - Find the value of a for which the line through the...Ch. 1.2 - 86. Rework Exercise 85, where the word parallel is...Ch. 1.2 - Prove the parallel property. [Hint: If...Ch. 1.2 - Prob. 88ECh. 1.2 - Prob. 89ECh. 1.2 - Prob. 90ECh. 1.2 - 91. Costs and Revenue A T-shirt company has fixed...Ch. 1.2 - 92. Costs and Revenue A company produces a single...Ch. 1.2 - Demand and Revenue Suppose that the quantity q of...Ch. 1.2 - 94. Demand and Revenue Suppose that the number n...Ch. 1.2 - Setting a Price During 2015, a manufacturer...Ch. 1.2 - 96. Setting a Price Rework Exercise 95 with a 2015...Ch. 1.2 - 97. Graph the three lines together, and then...Ch. 1.2 - Prob. 98ECh. 1.2 - Prob. 99ECh. 1.2 - Prob. 100ECh. 1.2 - Prob. 101ECh. 1.3 - Figure 9 shows a type of polygon that plays a...Ch. 1.3 - Figure 9 shows a type of polygon that plays a...Ch. 1.3 - Figure 9 shows a type of polygon that plays a...Ch. 1.3 - In Exercises 16, find the point of intersection of...Ch. 1.3 - In Exercises 16, find the point of intersection of...Ch. 1.3 - In Exercises 16, find the point of intersection of...Ch. 1.3 - In Exercises 1–6, find the point of intersection...Ch. 1.3 - In Exercises 16, find the point of intersection of...Ch. 1.3 - Prob. 6ECh. 1.3 - 7. Does satisfy the following system of linear...Ch. 1.3 - Does (12,4) satisfy the following system of linear...Ch. 1.3 - In Exercises 9-12, solve the systems of linear...Ch. 1.3 - In Exercises 9-12, solve the systems of linear...Ch. 1.3 - In Exercises 9-12, solve the systems of linear...Ch. 1.3 - In Exercises 9-12, solve the systems of linear...Ch. 1.3 - In Exercises 13–16, find the coordinates of the...Ch. 1.3 - In Exercises 13–16, find the coordinates of the...Ch. 1.3 - In Exercises 13–16, find the coordinates of the...Ch. 1.3 - In Exercises 1316, find the coordinates of the...Ch. 1.3 - Supply Curve The supply curve for a certain...Ch. 1.3 - 18. Demand Curve The demand curve for a certain...Ch. 1.3 - 19. Supply and Demand Suppose that supply and...Ch. 1.3 - Supply and Demand A discount book seller has...Ch. 1.3 - 21. Supply and Demand Suppose that the demand...Ch. 1.3 - Prob. 22ECh. 1.3 - Temperature Conversion The formula for converting...Ch. 1.3 - Prob. 24ECh. 1.3 - 25. Manufacturing A clothing store can purchase a...Ch. 1.3 - 26. Time Apportionment A plant supervisor must...Ch. 1.3 - Calling Card Options A calling card offers two...Ch. 1.3 - 28. Towing Fees Sun Towing Company charges $50...Ch. 1.3 - In Exercises 29 and 30, find the area of the...Ch. 1.3 - Prob. 30ECh. 1.3 - Prob. 31ECh. 1.3 - 32. Sales Determination An appliance store sells a...Ch. 1.3 - Prob. 33ECh. 1.3 - In Exercises 33-36, graph the lines and estimate...Ch. 1.3 - Prob. 35ECh. 1.3 - In Exercises 33-36, graph the lines and estimate...Ch. 1.4 - 1. Can a vertical distance be negative?
Ch. 1.4 - Under what condition will a vertical distance be...Ch. 1.4 - Suppose that the line y=3x+1 is used to fit the...Ch. 1.4 - 2. Suppose that the line is used to fit the four...Ch. 1.4 - 3. Find the sum-of-squares error E for the...Ch. 1.4 - Find the sum-of-squares error E for the...Ch. 1.4 - Complete Table 6, and find the values of m and b...Ch. 1.4 - 6. Complete Table 7, and find the values of m and...Ch. 1.4 - Consider the data points (1,2),(2,5),and(3,11)....Ch. 1.4 - 8. Consider the data points . Find the straight...Ch. 1.4 - 9. Consider the data points, and . Find the...Ch. 1.4 - Consider the data points (1,5),(2,7),(3,6), and...Ch. 1.4 - 11. Consider the data points and .
(a) Find the...Ch. 1.4 - Prob. 12ECh. 1.4 - According to Example 2, the sum-of-squares error...Ch. 1.4 - Prob. 14ECh. 1.4 - Fuel Economy The following table gives the city...Ch. 1.4 - 16. Pizzerias The following table gives the number...Ch. 1.4 - Lung cancer and Smoking The following table gives...Ch. 1.4 - Cigarette Use The percentage of college freshmen...Ch. 1.4 - 19. College Graduates Figure 15 gives the percent...Ch. 1.4 - Average College Costs Figure 16 gives the average...Ch. 1.4 - Life Expectancy The following table is an...Ch. 1.4 - Banking Two Harvard economists studied countries...Ch. 1.4 - 23. Consumer Price Index The following table gives...Ch. 1.4 - Greenhouse Gases Although greenhouse gases are...Ch. 1 - How do you determine the coordinates of a point in...Ch. 1 - 2. What is meant by the graph of an equation in x...Ch. 1 - 3. What is the y-intercept of a line? How do you...Ch. 1 - 4. What is the x-intercept of a line? How do you...Ch. 1 - Prob. 5FCCECh. 1 - Prob. 6FCCECh. 1 - Prob. 7FCCECh. 1 - Prob. 8FCCECh. 1 - Prob. 9FCCECh. 1 - Prob. 10FCCECh. 1 - Prob. 11FCCECh. 1 - What can you say about the slopes of perpendicular...Ch. 1 - Prob. 13FCCECh. 1 - Prob. 14FCCECh. 1 - What is the least-squares line approximation to a...Ch. 1 - 1. What is the equation of the y-axis?
Ch. 1 - Prob. 2RECh. 1 - Find the point of intersection of the pair of...Ch. 1 - Prob. 4RECh. 1 - Prob. 5RECh. 1 - Prob. 6RECh. 1 - Prob. 7RECh. 1 - Find the y-coordinate of the point having...Ch. 1 - Prob. 9RECh. 1 - Solve the system of linear equations....Ch. 1 - Prob. 11RECh. 1 - Cost of Moving The fee charged by a local moving...Ch. 1 - Are the points (1,2),(2,0), and (3,1) on the same...Ch. 1 - Prob. 14RECh. 1 - Prob. 15RECh. 1 - Prob. 16RECh. 1 - Prob. 17RECh. 1 - Prob. 18RECh. 1 - Prob. 19RECh. 1 - Prob. 20RECh. 1 - Prob. 21RECh. 1 - 22. Supply and Demand Consider the following four...Ch. 1 - Prob. 23RECh. 1 - Prob. 24RECh. 1 - Car Rentals One-day car rentals cost $50 plus 10...Ch. 1 - 26. Inflation In 2003, bacon cost an average of...Ch. 1 - Medical assistant Jobs According to the U.S....Ch. 1 - Prob. 28RECh. 1 - Prob. 29RECh. 1 - Prob. 30RECh. 1 - Prob. 31RECh. 1 - 32. Life Expectancy The following table gives the...Ch. 1 - Prob. 33RECh. 1 - Prob. 34RECh. 1 - Prob. 35RECh. 1 - Prob. 36RECh. 1 - 37. When is the x-intercept of a line the same as...Ch. 1 - 38. Suppose that you have found the least-squares...Ch. 1 - Does every line have an x-intercept? A...Ch. 1 - 40. Consider the line having the slope–intercept...Ch. 1 - Break-Even Analysis
We discussed linear demand...Ch. 1 - Break-Even Analysis We discussed linear demand...Ch. 1 - Break-Even Analysis
We discussed linear demand...Ch. 1 - Break-Even Analysis We discussed linear demand...Ch. 1 - Break-Even Analysis
We discussed linear demand...Ch. 1 - Break-Even Analysis
We discussed linear demand...Ch. 1 - Break-Even Analysis We discussed linear demand...Ch. 1 - Break-Even Analysis
We discussed linear demand...
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- Total marks 16 5. Let (N,F,P) be a probability space and let X : N → R be a random variable such that the probability density function is given by f(x)=ex for x € R. (i) Find the characteristic function of the random variable X. [8 Marks] (ii) Using the result of (i), calculate the first two moments of the random variable X, i.e., E(X") for n = 1,2. (iii) What is the variance of X. [6 Marks] [2 Marks]arrow_forward6. Let P be the standard normal distribution, i.e., P is the proba- bility measure on (R, B(R)) given by 1 dP(x) = 를 = e dx. √2πT Consider the random variables 21 fn(x) = (1 + x²) en+2, x Є R, n Є N. Using the dominated convergence theorem, prove that the limit Total marks 9 exists and find it. lim E(fn) n∞ [9 Marks]arrow_forwardRefer to page 38 for solving an optimal control problem using dynamic programming. Instructions: • Define the value function and derive the Hamilton-Jacobi-Bellman (HJB) equation. • Solve the HJB equation explicitly, showing all intermediate steps and justifications. Verify the solution satisfies the boundary conditions and optimality. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 18 for solving a second-order linear non-homogeneous differential equation. Instructions: Solve the associated homogeneous equation first. Use either the method of undetermined coefficients or variation of parameters for the particular solution. • Provide detailed steps for combining solutions into the general solution. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward6. Let X be a random variable taking values in (0,∞) with proba- bility density function fx(u) = 5e5u u > 0. Total marks 8 Let Y = X2. Find the probability density function of Y. [8 Marks]arrow_forward5. Let a probability measure P on ([0,3], B([0,3])) be given by 1 dP(s): = ½ s² ds. 9 Consider a random variable X : [0,3] → R given by X(s) = s², sc [0,3]. S Total marks 7 Find the distribution of X. [7 Marks]arrow_forward
- Refer to page 24 for solving a differential equation using Laplace transforms. Instructions: Take the Laplace transform of the given equation, applying initial conditions appropriately. ⚫ Solve the resulting algebraic equation and find the inverse transform. Provide step-by-step solutions with intermediate results and final verification. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 30 for deriving the Euler-Lagrange equation for an optimal control problem. Instructions: • Use the calculus of variations to derive the Euler-Lagrange equation. Clearly define the functional being minimized or maximized. Provide step-by-step derivations, including all necessary boundary conditions. Avoid skipping critical explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 32 for solving a linear-quadratic regulator (LQR) problem. Instructions: • Formulate the cost functional and state-space representation. • Derive the Riccati equation and solve it step-by-step. Clearly explain how the optimal control law is obtained. Ensure all matrix algebra is shown in detail. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 14 for solving a linear first-order differential equation. Instructions: • Convert the equation into its standard linear form. • Use integrating factors to find the solution. Show all steps explicitly, from finding the factor to integrating and simplifying the solution. Link [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 10 for a problem involving solving an exact differential equation. Instructions: • Verify if the equation is exact by testing әм მყ - ƏN მე If not exact, determine an integrating factor to make it exact. • Solve step-by-step, showing all derivations. Avoid irrelevant explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Haz b9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 10 for a problem involving solving an exact differential equation. Instructions: Verify exactness carefully. ⚫ If the equation is not exact, find an integrating factor to make it exact. Solve step-by-step and ensure no algebraic steps are skipped. Provide detailed explanations for each transformation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
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