# Solving linear equations - two views

Here are two views of symbolically representing two linear functions plotted in the {x,y} plane. 1) A pair of simultaneous linear equations of the form ax + by = c and Ax + By = C, or 2) A single equation of the form px + q = Px + Q. If you regard the two linear functions as a pair of simultaneous linear equations then the solution set is the point in the {x,y} plane where the graphs of ax + by = c and Ax + By = C intersect. [what is the significance of the value(s) of y in the solution point {x,y}? How many such values are there? Why?] If you regard the two linear functions as a single equation of the form px + q = Px + Q then the solution set is the point on the x axis whose x coordinate is the x coordinate of the intersection point of the two graphs. Which formulation do you prefer ? Why ? Does this perspective extend to functions other than linear - i.e., F(x) = G(x) as opposed to y = F(x) and y = G(x) ? Does this perspective extend to functions of more than one variable - i.e., F(x,y) = G(x,y) as opposed to z = F(x,y) and z = G(x,y) ? What problems could/would you pose to your students based on this applet ?
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