Recently, I saw a video in which a teenage girl was putting on makeup and questioning why higher math was as focused on as it was in antiquity. She specifically mentioned the Pythagorean theorem, and then mentioned slope-intercept form, y = mx +b. I knew why Pythagoras was interested in math such that playing with squares until you find the ratios of the sides of a right triangle would be considered a worthy philosophical pursuit, but slope-intercept form stumped me. I genuinely had no idea who the first person to write y = mx + b was.
Given that Y and X are used to express points on a line, I know that it has to come post Descartes creating the field of linear algebra, and research online indicated that Euler may have been the more common mathematician to use m as a stand-in for slope. But in my research, I couldn’t find a particular attestation to a specific work by either mathematician, (or any other, for that matter) which indicated who actually figured out that a linear equation could be expressed this way. So my question is, which mathematician (of these two or beyond) first formulated the slope-intercept equation, and in what work did they formulate it?
The first use of y = mx + b appears to be in Salmon's Conic Sections:
George Salmon was an Irish mathematician who worked at Trinity College as a mathematician and theologian. The first edition of his book was published in 1848, but the oldest I can check for y=mx+b is the 3rd edition of 1855 linked above.
However, the other common modern version, y = mx + c, is older, appearing in O'Brien's A Treatise on Plane Co-ordinate Geometry on 1844:
Matthew O'Brien was also an Irish mathematician, at Caius College (University of Cambridge) at the time.
There might be older versions, but the detailed geometric derivation in these books suggests that this algebraic expression of a straight line was relatively new at the time.
Reference:
A brief history of y = mx + b, attributed to J. Miller (the reference is a dead link) is given at Weisstein, Eric W. "Slope." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Slope.html (although only noting O'Brien's use of m rather than y=mx+c).