Systems
of NonLinear Equations:
y
= x^{2} From the form of the equations, you should know that this system contains a parabola and a circle.
From the first equation, I think I'll plug in "y" for "x^{2}" in the second equation, and solve: x^{2}
+ (y – 2)^{2} = 4 Now I need to find the corresponding xvalues. When y = 0: y = x^{2}0 = x^{2}0 = x (This is the solution at the origin that we'd been expecting.) When y = 3: y = x^{2}3 = x^{2}±sqrt(3) = x Then the solutions are the points , (0, 0), and .
3x^{2}
+ 2y^{2} = 35 I can rearrange the first equation to get: This tells me that the first equation is an ellipse. However, rather than graphing this using ellipse formulae, you could also solve to get a "plusminus" expression that you can graph as two equations: Copyright © 20022011 Elizabeth Stapel All Rights Reserved (You would plug this into your graphing calculator as two graphs, one graph for the top "plus" part of the ellipse, and another for the bottom "minus" part.) The second equation rearranges as: ...which is an hyperbola. The second equation also solves (for your graphing calculator) as: Whatever format you use (the ellipse and the hyperbola centervertex forms, or the "plusminus" forcalculator forms), this system graphs as: As you can see, there appear to be four solutions. To find them algebraically, I will choose to solve the second equation for x^{2} (rather than just x), and plug the resulting expression into the first equation, which I will then solve for y:. (It's okay that I "only" solve for x^{2}, because neither equation has an xterm. There is no need, in this particular case, to do any more solving.) 4x^{2}
– 3y^{2} = 24 Then, subsituting into the first equation for the x^{2}, I get: 3x^{2}
+ 2y^{2} = 35
When y = –2: x^{2} = (^{ 3}/_{4} )y^{2} + 6 = (^{ 3}/_{4} )(–2)^{2} + 6 = (^{ 3}/_{4} )(4) + 6 = 3 + 6 = 9 x = ± 3 When y = 2: x^{2} = (^{ 3}/_{4} )y^{2} + 6 = (^{ 3}/_{4} )(2)^{2} + 6 = (^{ 3}/_{4} )(4) + 6 = 3 + 6 = 9 x = ± 3 Then the solution is the points (–3, –2), (–3, 2), (3, –2), and (3, 2), which may also be written as (± 3, ± 2), since all the "plusminus" combinations are included. << Previous Top  1  2  3  4  5  6  Return to Index Next >>



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