# Help with this math problem

**URL:** <https://boards.straightdope.com/t/help-with-this-math-problem/1013209>\
**Category:** Factual Questions\
**Tags:** science-math\
**Created:** [January 21, 2025, 1:05pm UTC](https://boards.straightdope.com/t/help-with-this-math-problem/1013209 "2025-01-21T13:05:36Z")\
**Posts on this page:** 5\
**Page:** 2

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**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\
**Post date:** [January 22, 2025, 7:43pm UTC](https://boards.straightdope.com/t/help-with-this-math-problem/1013209/21 "2025-01-22T19:43:39Z")

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> [@Earl\_Snake-Hips\_Tucker](#):
>
> From. . . “Fundamentals of Algebra and Trigonometry” by Earl Swokowski. Second edition. 1971. #20 on Page 51 🙂

OK now that makes sense. The OP’s question needs to be looked at in relation to all of the other possibilities the question asks about.

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**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [January 22, 2025, 8:06pm UTC](https://boards.straightdope.com/t/help-with-this-math-problem/1013209/22 "2025-01-22T20:06:21Z")

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If you have a linear equation ax=b, what does the space of solutions look like? It could be empty, or it could be an affine space. In one dimension, there are not too many possibilities: empty, a single point, or a 1-dimensional space (which is then the whole real line).

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**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\
**Post date:** [January 23, 2025, 8:50am UTC](https://boards.straightdope.com/t/help-with-this-math-problem/1013209/23 "2025-01-23T08:50:55Z")

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Back in high school we would have most certainly have taken the _solution set_ to mean the set of all solutions as defined by an appropriate algebraic manipulation.

I think we did basic slope intercept in first year high school, so year 8. Essentially very introductory algebra. Very quickly leading into the classic problems represented by line intersections.

The use of the idea of a _set_ of solutions might partially be due to the introduction of naive set theory as part of New Maths. As little kids we learnt all about set theory with Venn Diagrams and learnt about Cardinal numbers aka counting numbers. So getting enthusiastic about solutions as _sets_ is hardly a surprise.

Intensional notation seemed to just leak into one’s consciousness, and I would be hard pressed to say when that started.

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**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\
**Post date:** [January 23, 2025, 12:17pm UTC](https://boards.straightdope.com/t/help-with-this-math-problem/1013209/24 "2025-01-23T12:17:54Z")

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> [@Francis\_Vaughan](#):
>
> The use of the idea of a _set_ of solutions might partially be due to the introduction of naive set theory as part of New Maths.

I suspect that may be when/why it gained traction. Textbooks that are being precise still talk about the _solution set_, rather than just the _solution_, of an equation to avoid the implication that there’s necessarily one and only one solution.

If you look at the image taken directly from the book in question, linked to in Post #16, you’ll see that they explicitly mention the possibility that the solution set could be the empty set, or the set of all real numbers. (And in the subsequent problem they refer to the concept of _equivalent equations_, which are equations that have the same solution set as each other.)

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**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\
**Post date:** [January 27, 2025, 3:44am UTC](https://boards.straightdope.com/t/help-with-this-math-problem/1013209/25 "2025-01-27T03:44:28Z")

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Thanks, everyone.

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