I have just started reading basic algebra and I have this curiosity that came up when solving basic linear equations. Some equations have no solutions. Are there any real world example of equations with no solutions?
For example consider the equation: $4x -24 = 4x - 32$. It has no solution. Do we find such cases in real world?
Please ignore any naiveness in the question as I have just started off in mathematics and do not have a deep understanding of it. Thanks.
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1$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotCommented Mar 23, 2023 at 17:24
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$\begingroup$ It depends on what "real world" means. $\endgroup$– RandallCommented Mar 23, 2023 at 17:34
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3$\begingroup$ It happens when two quantities change according to linear functions with the same slope/rate of change. For example, determine time when a clock in London will show the same time as a clock in New York if the time difference is 5 hours. $\endgroup$– VasiliCommented Mar 23, 2023 at 19:07
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$\begingroup$ Lookup Fermat's Last Theorem. $\endgroup$– dxivCommented Mar 24, 2023 at 2:32
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2$\begingroup$ $x = x + 1$ "The number of apples I have is one more than the number of apples I have" $\endgroup$– BlueRaja - Danny PflughoeftCommented Mar 24, 2023 at 5:41
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2 Answers
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Sure. You and I run a race, but you get a five second head start over me. We both run at the same constant speed of 10 feet per second. At what time will we cross paths during the race?
The distance you cover (in feet) expressed in terms of time $t\geq 0$ elapsed (in seconds) since you started is $10t.$ The distance I cover is $0$ for $0\leq t\leq 5$ and $10(t-5)$ for $t\geq 5.$ There is no $t> 0$ where our distances coincide: we will never cross paths during the race.
answered Mar 23, 2023 at 17:37
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I see a few solutions of the form $mx = mx + c$, which is very clear: two parallel lines never cross. But I wanted to add that even if the lines cross, their intersection may not be a valid solution because of other constraints.
For example, suppose I run a company that makes widgets. Each widget I make costs me \$100, and I can sell any number of widgets for \$95 each. How many widgets should I make and sell to earn a profit of \$1,000,000?
You can turn this into an equation, $1000000=95w-100w$, and solve it to discover that I can achieve my goal by making -200,000 widgets. Fine, but of course I can't actually do that in real life. The equation has a solution in the domain of integers, but we're really working with the non-negative integers, where it has no solution.
