Systems
of NonLinear Equations: To find the exact solution to a system of equations, you must use algebra. Let's look at that first system again:
y
= x^{2} Since I am looking for the intersection points, I am therefore looking for the points where the equations overlap, where they share the same values. That is, I am trying to find any spots where y = x^{2} equals y = 8 x^{2}: y = x^{2} = y = 8 x^{2} The algebra comes in when I manipulate useful bits of this last equation. I can pick out whichever parts I like. (They're all equal, after all  at least at the intersection points, but the intersection points are the only points that I care about anyway!) So I can pick out any of the following: y
= x^{2}y = 8 x^{2}y
= y Each of these subequations is true, but only the last one is usefully new and different: x^{2} = 8 x^{2} I can solve this for the xvalues that make the equation true: x^{2}
= 8 x^{2}2x^{2} = 8 Then the solutions to the original system will occur when x = 2 and when x = +2. What are the corresponding yvalues? To find them, I plug the xvalues back in to either of the two original equations. (It doesn't matter which one I pick because I only care about the points where the equations spit out the same values. So I can pick whichever equation I like better.) I'll plug the xvalues into the first equation, because it's the simpler of the two: x = 2: y = x^{2}y = (2)^{2} = 4 x = +2: y = x^{2}y = (+2)^{2} = 4 Then the solutions (as we already knew) are (x, y) = (2, 4) and (2, 4). In the above example, the solutions were nice, "neat" integer values; no fractions or decimals. But solutions will not always be neat, so, while the pictures can be very useful for giving you a "feel" for what is going on, graphing is not as accurate as doing the algebra. Warning: Students are often taught nowadays to "round" absolutely everything, and are thus implicitly taught that all answers will be "neat" answers. But this is wrong; don't fall for it! For instance:
y
= x^{2} + 3x + 2 I can solve this in the same manner as we did on the previous problem. The "solution" to the system will be any point(s) that the lines share; that is, any point(s) where the xvalue and corresponding yvalue for y = x^{2} + 3x + 2 is the same as the xvalue and corresponding yvalue for y = 2x + 3; that is, where the lines overlap or intersect; that is, where y = x^{2} + 3x + 2 equals y = 2x + 3. Copyright 20022011 Elizabeth Stapel All Rights Reserved
For purposes of graphing, the approximate solutions are: (x, y) = (1.62, 0.24) and (0.62, 4.24). In other words, while my guess from the picture was close, it was not entirely correct nor "exact". (Note: If the algebra had given me answers that are far afield of these picturebased guesses, I would have been able to safely assume that I had messed up the math somewhere. In this way, the graph can be very helpful for checking your work.) << Previous Top  1  2  3  4  5  6  Return to Index Next >>


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