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Systems of Equations and Inequalities

See the attached file.

In exercise 44 and 46, let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers.

44) The sum of two numbers is 2. If one number is subtracted from the other, their difference is 8. Find the numbers.

46) The sum of three times a first number and twice a second number is 8. If the second number is subtracted from twice the first number, the result is 3. Find the numbers.

Exercises 62 and 64 describe a number of business ventures. For each exercise,
a. Write the cost function, C.
b. Write the revenue function, R.
c. Determine the break-even point. Describe what this means.

62) A company that manufactures bicycles has a fixed cost of $1000, 000. It costs $100 to produce each bicycle. The selling price is $300 per bike. (In solving this exercise, let x represent the number of bicycles produced and sold.)

64) You invested $30,000 and started a business writing greeting cards. Supplies cost 2 cent per card and you are selling each card for 50 cent. (In solving this exercise, let x represent the number of cards produced and sold.)

68) One of the most dramatic developments in the work force has been the increase in the number of women, at approximately ½ % per year. By contrast, the percentage of men is decreasing by ¼% per year. The graphs shown below illustrate these changes.

The function y = 0.52x + 35.7 models the percentage, y, of U.S. women in the work force x years after 1955. The function 0.25x + y = 85.4 models the percentage, y, of U.S. men in the work force x years after 1955. Use these models to determine when the percentage of women in the work force will be the same as the percentage of men in the work force. Round to the nearest year. What percentage of women and what percentage of men will be in the work force at that time?

70) The graph indicates that in 1984, there were 72 meals per person at take-out restaurants. For the period shown, this number increased by an average of 2.25 meals per person per year. In 1984, there were 94 meals per person at on-premise dining facilities and this number decreased by an average of 0.55 meals per person per year.

a. Write a function that models the average number of meals per person at take-out restaurants x years after 1984.
b. Write a function that models the average number of meals per person at on-premise dinning facilities x years after 1984.
c. In which year, to the nearest whole year, was the average number of meals per person for take-out and on-premise restaurants the same? For that year, how many meals per person, to the nearest whole number, were there for each kind of restaurant? Which kind of restaurant had the greater number of meals per person after that year?

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Solution Preview

Please see attached for the solution to the posted problems.

In exercise 44 and 46, let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers.

44) The sum of two numbers is 2. If one number is subtracted from the other, their difference is 8. Find the numbers.
Solution: We'll have two equations:

Using elimination, we have:

Solving for y, we have:

46) The sum of three times a first number and twice a second number is 8. If the second number is subtracted from twice the first number, the result is 3. Find the numbers.
Solution: We'll have two equations:

Using elimination, we have:

Solving for y, we have:

Exercises 62 and 64 describe a number of business ventures. For each exercise,
a. Write the cost function, C.
b. Write the revenue function, R.
c. Determine the break-even point. Describe what this means.

62) A company that manufactures bicycles has a fixed cost ...

Solution Summary

The solution contains the solution to the given system of equations and inequalities problems.

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