In most math curricula, the solving of equations and inequalities is limited to linear and quadratic examples. Here are two views of solving such equations - There are two views about how to symbolically describe two linear functions plotted in the {x,y} plane - either as a pair of simultaneous equations of the form y=f(x) and y=g(x), or as a single equation of the form f(x)=g(x).
1) A pair of simultaneous linear equations of the form ax + by = c and Ax + By = C. 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. 2) A single equation of the form px + q = Px + Q. 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.
A similar set of remarks can be made about solving quadratic equations. The solution set consists either of the set of points in the {x,y} plane where the two quadratics intersect, or the points on the x
axis whose x coordinates are the x coordinates of the intersection points
of the two graphs. This distinction is of particular importance if one is using these equations to model entities in the world in which we live. [See essay on Semantic Aspects of Quantity]. You can explore these two ways of thinking about linear equations here. You can explore these two way of thinking about quadratic equations here. These applets allow you to solve linear & quadratic equations & inequalities by manipulating their graphs OR by manipulating the symbolic expressions of the functions being compared These applets allow you to UNsolve the solution sets of linear & quadratic equations & inequalities in order to explore the equivalence classes of equations & inequalities with those solution sets |