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 this case, the solutions were "neat" 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 our guess from the picture was close, it was not entirely correct. (However, 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 helpful for checking your work.) << Previous Top  1  2  3  4  5  6  Return to Index Next >>



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