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# Algebra

## Graphing Algebraic Equations

The methods of algebra are extended to geometry, and vice versa, by graphing. The value of graphing is two-fold. It can be used to describe geometric figures using the language of algebra, and it can be used to depict geometrically the algebraic relationship between two variables. For example, suppose that Fred is twice the age of his sister Irma. Since Irma's age is unknown, Fred's age is also unknown. The relationship between their ages can be expressed algebraically, though, by letting y represent Fred's age and x represent Irma's age. The result is y = 2x. Then, a graph, or picture, of the relationship can be drawn by indicating the points (x,y) in the Cartesian coordinate system for which the relationship y = 2x is always true. This is a straight line, and every point on it represents a possible combination of ages for Fred and Irma (of course negative ages have no meaning so x and y can only take on positive values). If a second relationship between their ages is given, for instance, Fred is three years older than Irma, then a second equation can be written, y = x+3, and a second graph can be drawn consisting of the ordered pairs (x,y) such that the relationship y = x+3 is always true. This second graph is also a straight line, and the point at which it intersects the line y = 2x is the point corresponding to the actual ages of Irma and Fred. For this example, the point is (3,6), meaning that Irma is three years old and Fred is six years old.