a. Explanation Given: A woman has $1,400,000 to purchase three investments. One investment earns 8%, a second earns 7.5%, and third earns 10%. Method used: Left-to-Right Elimination Method (or Back substitution): 1. If necessary, interchange two equations or use multiplication to make the coefficient of x in the first equation a 1. 2. Add a multiple of the first equation to each of the following equations so that the coefficients of x in second and third equations become 0. 3. Multiply (or divide) both sides of the second equation by a number that makes the coefficient of y in the second equation equal to 1. 4. Add a multiple of the (new) second equation to the (new) third equation so that the coefficient of y in the newest third equation becomes 0. 5. Multiply (or divide) both sides of the third equation by a number that makes the coefficient of z in the third equation equal to 1. This gives the solution for z in the system of equations. 6. Use the solution for z to solve y in the second equation. Then substitute values for y and z to solve for x in the first equation. (This is called back substitution) Calculation: Let x be the amount of first investment that pays 8%, y be the amount of second investment that pays 7.5%, and z be the amount of third investment that pays 10% Hence, the equation x + y + z = 1 , 400 , 000 , represents the sum of three investments, since the investor invests $1,400,000. Similarly, the 0.08 x + 0.075 y + 0.10 z = 120 , 000 represents the sum of total interest earned after investing $120,000 in three different investments that pay 8%, 7.5%, and 10%. Combine the above two equation and form the system of two equations in three variables as follows. { x + y + z = 1 , 400 , 000 0.08 x + 0.075 y + 0.10 z = 120 , 000 Use the method stated above and solve the system of equations for x , y , and z as follows. Perform the row operations R 2 → R 2 − 0.08 R 1 and obtain the new equivalent system of equations as follows. { x + y + z = 1 , 400 , 000 − 0.005 y + 0.02 z = 8000 From the above system observe that the variable z is not a leading variable in any of the two equations. Hence, the variables x and y can be solve in terms of z as follows. The second equation gives, − 0.005 y + 0.02 z = 8000 − 0.005 y = 8000 − 0.02 z y = − 1 0.005 ( 8000 − 0 b. To determine The investment strategy that will minimize her risk and return $120,000 if the 10% investment has the most risk.

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Chapter 7.1, Problem 40E

To determine

The strategies that gives the annual earnings of $120,000 from the three investments.

To determine

The investment strategy that will minimize her risk and return $120,000 if the 10% investment has the most risk.

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Chapter 7.1, Problem 39E

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1. 2. Show that the following are not logically equivalent by finding a counterexample: (p^q) →r and (db) V (d←d) Show that the following is not a contradiction by finding a counterexample: (pV-q) AqA (pv¬q Vr) 3. Here is a purported proof that (pq) ^ (q → p) = F: (db) v (bd) = (db) v (bd) =(qVp) A (g→p) = (¬¬q V ¬p) ^ (q→ p) (db) V (db) = =¬(a→p)^(a→p) = (gp) ^¬(a → p) =F (a) Show that (pq) ^ (q→p) and F are not logically equivalent by finding a counterex- ample. (b) Identify the error(s) in this proof and justify why they are errors. Justify the other steps with their corresponding laws of propositional logic.

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MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- for College Algebra in Context

Ch. 7.1 - In Exercises 14, solve the systems of equations....Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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The strategies that gives the annual earnings of $120,000 from the three investments.

Solution Summary: The author explains the Left-to-Right Elimination Method (or Back substitution) for calculating the annual earnings of 120,000 from three investments.

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The strategies that gives the annual earnings of $120,000 from the three investments.

The investment strategy that will minimize her risk and return $120,000 if the 10% investment has the most risk.

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