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The Cartesian coordinate system is what I have been told provided the first link between algebra and geometry. However, I also have learned that, for instance, Omar Khayyam solved cubic equations as the intersection between parabolas.

So, how was this visualization done before Descartes? Also, what is so fundamentally different between the Cartesian coordinate system and the older methods?

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Cartesian coordinates provided the first systematic way of converting geometric problems into algebraic ones and vice versa, but one can do that in elementary geometry without any coordinates simply by denoting sides and angles of polygons by letters and using trigonometry. This is more or less what ancient astronomers did when computing with their geometric models after the father of astronomy Hipparchus introduced chords, the predecessors of sine and cosine, in 2nd century BC. This approach was generalized and perfected by the founder of modern symbolic algebra Viete, who already used modern trigonometry, even before Descartes. See Viète's Relevance and his Connection to Euler.

The use of geometry to solve (what we would call) polynomial equations is also ancient. Omar Khayyam was following in the footsteps of Menaechmus, Archimedes and Diocles, the first of whom lived before Euclid. Of course, ancient geometers did not start with equations, usually the initial problem was itself geometric, e.g. Menaechmus used intersection of curves to duplicate the cube, which in algebraic terms translates into solving $x^3=2$. Once Islamic mathematicians developed algebraic notation however, starting with al-Khwarismi in the 9th century AD, they started converting geometric problems into equations and back, and using ancient geometric methods to solve them. See How was geometry historically used to solve polynomial equations?.

In 3rd century BC Apollonius of Perga already used predecessors of Cartesian coordinates to study conic sections, but again he used them to solve geometric problems since there was no algebra then. Only in hidsight can we interpret him as visualizing algebra. Oresme used primitive coordinate graphs to visualize motion during middle ages, the algebra was in diapers back then. Both of them influenced Descartes, see When do we see for the first time the use of the Cartesian plane?

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