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How can the slope and one point on a line be used to write its equation?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, you can determine the equation of a straight line using its slope and a single known point by utilizing the point-slope form.1

If you have a line with a slope $m$ that passes through a specific point $(x_{1}, y_{1})$ , the equation is expressed as:

$y - y_{1} = m (x - x_{1})$

In this equation, $m$ represents the constant rate of change (slope) of the line, while $(x_{1}, y_{1})$ are the coordinates of the given point. By substituting these known values into the formula, you create an algebraic relationship that defines every point $(x, y)$ on that line. This method is a fundamental application of the algebraic techniques developed by mathematicians like René Descartes and Pierre de Fermat to describe geometric figures.12