Question

You're a manager in a company that produces rocket ships. Machine
\[\text{A}\] and Machine
\[\text{B}\] both produce cockpits and propulsion systems. Machine
\[\text{A}\] and Machine
\[\text{B}\] produce cockpits at the same rate, and they produce propulsion systems at the same rate. Machine
\[\text{A}\] ran for
\[26\] hours and produced
\[4\] cockpits and
\[6\] propulsion systems. Machine
\[\text{B}\] 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?

Answer 1

No — the data are inconsistent with the assumption that both machines share the same production rates, so there is no unique solution.

Let x = cockpits produced per hour (same for both machines) and y = propulsion systems per hour (same for both). From Machine A:
26x = 4 → x = 4/26 = 2/13 ≈ 0.15385 per hour,
26y = 6 → y = 6/26 = 3/13 ≈ 0.23077 per hour.

From Machine B:
56x = 8 → x = 8/56 = 1/7 ≈ 0.14286 per hour,
56y = 12 → y = 12/56 = 3/14 ≈ 0.21429 per hour.

Because 2/13 ≠ 1/7 (and 3/13 ≠ 3/14), the same (x,y) cannot satisfy both machines' data. So the system is inconsistent — no common rates exist.

(If instead you drop the “same rate” assumption and let each machine have its own rates, you can compute each machine’s rates separately: A produces a cockpit in 26/4 = 6.5 hours and a propulsion in 26/6 ≈ 4.333 hours; B produces a cockpit in 56/8 = 7 hours and a propulsion in 56/12 ≈ 4.667 hours.)