Question

Please write a few sentences on the following two topics:

1. The system of coordinates is a "bridge" between algebra and geometry. Why? How? Etc....

2. Relations between straight lines and linear equations.

Please write neatly, and in detail. Thank you so much!

Answer 1

1. Rene Descartes, a French mathematician established the concept of co-ordinate geometry in the early 17th century. As per Rene Descartes’s work, points in a plane can be expressed in terms of two co-ordinates: x (measuring in one direction), and y (in another direction . However, these coordinates were not rectangular. The concept of rectangular coordinates has its origin in Egypt. The word 'Cartesian' is derived from Descartes' name.

As per Rene Descartes, the location of a point (x, y) in the x-y plane can be fixed , starting from a fixed point (the origin) by measuring its x-distance and y-distance from the origin.

This provides a link with algebra, since x and y are algebraic symbols used to denote independent and dependent numbers/variables respectively.

In late 16^th century, a Norman mathematician Nicole Oresme described a way of graphing linear relationships between an independent and a dependent variable. Finally, towards early 17^th century, Fermat and Descartes gave an insight into the procedure linking Algebra and Geometry.

2. A linear relationship means that is a relationship between two sets of variables which can be represented by a line .A linear equation in two variables describes a relationship in which the value of one of the variables (dependent variable) depends on the value of the other variable(independent variable). A linear equation is one way of characterizing the set of points which make up a given line. The equations are called linear because their graphs are straight lines. To describe a particular line, we need to specify two distinct pieces of information relating to that line. A specific straight line can be determined by specifying two distinct points that the line passes through, or it can be determined by giving one point that it passes through and describing the slope ( inclination to the X-Axis) of the line. The point-slope form of the equation of a line passing through the point (x₁,y₁) is y-y₁ = m(x-x₁) where m is the slope of the line. The slope-intercept form of the equation of a line is y = mx+c , where m is the slope of the line and c is the y-intercept ( distance from the origin, of the point where the line meets the Y-Axis).