ELIMINATION AND GRAPHING Systems of Nonlinear Equations

Nonlinear Systems Last unit we focused on solving systems of linear equations, now it is time to switch gears. We are now going to look at systems of nonlinear equations. We will have parabolas and lines, parabolas and circles, and parabolas and parabolas. We could have no solutions, one solution, or many solutions. Just like linear systems, there are 3 ways to solve these:  Elimination  Substitution  Graphing

Elimination Remember in elimination you are adding the 2 equations together to get rid of a variable.  2x – y = 3  x 2 – y = 2  -2x + y = -3  x 2 – y = 2  x 2 – 2x = -1  x 2 – 2x + 1 = 0  (x – 1)(x – 1) = 0 x = 1 2(1) – y = 3 -y = 1 y = -1 (1, -1) What variable should we get rid of? There is only one! Which equation is easier to multiply? I don’t want a negative “x 2 ”. Add the two equations. Move everything to one side so you can factor. Factor. Solve. This has only one solution, so the two equations only intersect once. Find “y”.

Elimination I know that seems hard, so lets try another one.  x 2 + y 2 = 1  x 2 + y = -1  x 2 + y 2 = 1  -x 2 – y = 1  y 2 – y = 2  y 2 – y – 2 = 0  (y – 2)(y + 1) = 0  y = 2 y = -1  x = -1x 2 – 1 = -1  x 2 = - 3x 2 = 0  Not realx = 0  (0, - 1) Start just like before. Solve. Plug the “y” values back in. You can’t take the square root of a negative number, so there is only one solution.

Elimination Sometimes, the equations will cross in 2 or 4 places.  -x + y = 3  x 2 – y = -1  x 2 – x = 2  x 2 – x – 2 = 0  (x – 2)(x + 1) = 0  x = 2 x = -1  -(2) + y = 3 -(-1) + y = 3  y = 5y = 2  (2, 5)(-1, 2) This is an equation of a line and a parabola. They might look like this if you graphed them:

Elimination What are these equations for? Parabola Circle Remember, when you take the square root of a number there are 2 answers.

Graphing Let’s switch gears and try a different method. Solving by graphing is very similar to what we did with linear equations. You need to transfer the graphs to paper and you have to follow the “intersection” steps every time the graphs intersect.  y = -x 2 – 3 and y = x – 5  (-2, 7) and (1, -4)