X-Intercept

The x-intercept is the name given to the intersection point(s) of the function/line with the x-axis (abscissa) for the ordinate value $ y = 0 $ (origin).

It is also the distance between the origin of the coordinate system (the coordinate point $ (0,0) $) and the point of intersection of the line/curve with the abscissa axis.

— With the equation

For any function (not necessarily affine/linear), calculating the x intercept is calculating the value for $ y = 0 $ or $ f(x) = 0 $. The value(s) obtained is/are the x-intercept(s).

For an equation of a line (in the 2D plane), the equation has the form $ y = a x + b $, finding $ y = 0 $ is solving $ a x + b = 0 $ so $ x = -b/a $

— With 2 points (line only)

Knowing 2 points belonging to the line, it is possible to find its equation (see the linear equation calculator) in order to deduce the x-intercept (see above)

— With the slope coefficient and 1 point (line only)

Knowing the slope coefficient of a line and one of the points belonging to it, it is possible to calculate the equation of this line (see the page on the calculation of linear equations) and therefore, with this equation, deduce the x-intercept (see above).

The x-intercept and the y-intercept are not the same thing.

The abscissa at the origin represents the point(s) of intersection of the value curve with the abscissa (x axis).

The ordinate at the origin represents the point of intersection of the curve with the ordinate (y axis).

Yes, there can be multiple values (or even an infinity), and there can also be none (not all functions necessarily cross the x-axis).