Saturday, July 25, 2020

Solving Systems of Linear Equations - Practice Problems and Answers (#4 Part 1)

Problems:

Solve the system of equations by any method:

8x + 16y = 200
60x + 40y = 960.

Solve the system of equations by any method:

5x + 2y = 54
2x + 4y = 60.

A company manufactures two types of printers, the Inkjet and Laser.  On the assembly line, the Inkjet requires 7 hours, while the Laser takes 11 hours.  The Inkjet requires one hour and the Laser requires 3 hours of testing.  On a particular production run the company has available 1,540 work hours on the assembly line and 360 work hours for testing.  Find the number of printers the company should make.

Answers:

Solve the system of equations by any method: 8x + 16y = 200 60x + 40y = 960. We will solve this system of equations by the substitution method. Note that 8x + 16y = 200 implies that 8x = 200 - 16y and moreover that x = 25 - 2y. Put x = 25 - 2y into 60x + 40y = 960 to get 60(25 - 2y) + 40y = 960 which implies that 1500 - 120y + 40y = 960 which implies that 1500 - 80y = 960 which implies that 540 - 80y = 0 which implies that 540 = 80y which implies that y = 27/4. Now put y = 27/4 into one of the equations in the original system. Without loss of generality, put y = 27/4 into 8x + 16y = 200 to get 8x + 16(27/4) = 200 implies that 8x + 108 = 200 implies that 8x = 92 implies that x = 23/2. Thus, the solution is (x, y) = (23/2, 27/4).

Solve the system of equations by any method: 5x + 2y = 54 2x + 4y = 60. We will solve this system of equations by the elimination method. Multiply 5x + 2y = 54 by -2 to get -10x -4y = -108. Add -10x - 4y = -108 to 2x + 4y = 60 to get -8x = -48 which implies that x = 6. Put x = 6 into 2x + 4y = 60 in the original system of equations (because we did not do anything to this equation) to get 2(6) + 4y = 60 which implies that 12 + 4y = 60 which implies that 4y = 48 which implies that y = 12. Thus, (x, y) = (6, 12) is the solution.

A company manufactures two types of printers, the Inkjet and Laser. On the assembly line, the Inkjet requires 7 hours, while the Laser takes 11 hours. The Inkjet requires one hour and the Laser requires 3 hours of testing. On a particular production run the company has available 1,540 work hours on the assembly line and 360 work hours for testing. Find the number of printers the company should make. Answer: Let I be the number of Inkjet printers and L be the number of Laser printers. We will solve the system 7I + 11L = 1540, I + 3L = 360. (The first equation was from the assembly line and the second equation was from testing.) Since I + 3L = 360, we have that I = 360 - 3L. Substitute I = 360 - 3L into 7I + 11L = 1540 to get 7(360 - 3L) + 11L = 1540 which implies that 2520 - 21L + 11L = 1540 which implies that 2520 - 10L = 1540 which implies that 980 - 10L = 0, i.e. 980 = 10L. Thus, L = 98. Put L = 98 into I + 3L = 360. (We could do 7I + 11L = 1540, but I + 3L = 360 is easier.) to get I + 3(98) = 360, i.e. I = 66. The company should make 98 Laser printers and 66 Inkjet printers.


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