You're a manager in a company that produces rocket ships. Machine A and Machine B both produce cockpits and propulsion systems. Machine A and Machine B produce cockpits at the same rate, and they produce propulsion systems at the same rate. Machine A ran for 26 hours and produced 4 cockpits and 6 propulsion systems. Machine B ran for 56 hours and produced 8 cockpits and 12 propulsion systems. We use a system of linear equations in two variables. Can we solve for a unique amount of time that it takes each machine to produce a cockpit and to produce a propulsion system? Choose 1 answer: (A) Yes; it takes Machine A and Machine B 2 hours to produce a cockpit and 3 hours to produce a propulsion system. (B) Yes; it takes Machine A. and Machine B 1 hour to produce a cockpit and 4 hours to produce a propulsion system. (C) No; the system has many solutions. (D) No; the system has no solution.

Question

You're a manager in a company that produces rocket ships. Machine A and Machine B both produce cockpits and propulsion systems. Machine A and Machine B produce cockpits at the same rate, and they produce propulsion systems at the same rate. Machine A ran for 26 hours and produced 4 cockpits and 6 propulsion systems. Machine  B ran for 56 hours and produced 8 cockpits and 12 propulsion systems. We use a system of linear equations in two variables. Can we solve for a unique amount of time that it takes each machine to produce a cockpit and to produce a propulsion system? Choose 1 answer: (A) Yes; it takes Machine A and Machine B 2 hours to produce a cockpit and 3 hours to produce a propulsion system. (B) Yes; it takes Machine A. and Machine B 1 hour to produce a cockpit and 4 hours to produce a propulsion system. (C) No; the system has many solutions. (D) No; the system has no solution.

Bytelearn · Accepted Answer

Denoting the production time: Let's denote the time it takes for Machine A or B to produce one cockpit as x hours and the time it takes to produce one propulsion system as y hours. Since Machine A and B produce at the same rate, we can set up two equations based on the information given: For Machine A: 4x + 6y = 26 (since it produced 4 cockpits and 6 propulsion systems in 26 hours) For Machine B: 8x + 12y = 56 (since it produced 8 cockpits and 12 propulsion systems in 56 hours)Setting up the equations: We notice that the second equation is exactly double the first equation. This means that the two equations are not independent; the second equation is a multiple of the first. Therefore, we cannot solve for unique values of x and y because we have one equation with two unknowns.Recognizing the relationship between equations: Since we cannot determine unique values for x and y, the correct answer is that the system has many solutions. This corresponds to option (C).

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