1. 5.5 Systems Involving Nonlinear Equations 1 In previous sections, we solved systems of linear equations using various techniques such as substitution, elimination. The solution to a system of two linear equations is: 1.A point (x,y) where the two lines intersected, 2.No solution if the lines are parallel, or 3.Infinitely many solutions if the lines are the same line. Case 1:Case 2:Case 3: one solution no solutioninfinitely many solutions Next Slide In this section we will look for the intersection of two equations where we may have graphs such as parabolas, circles, ellipses, hyperbolas and lines.

2. 5.5 Systems Involving Nonlinear Equations 2 Let’s look at an example of a parabola and an ellipse. How many points of intersection can exist? 4 points 3 points 1 points 2 points 0 points Next Slide One method of solving a system of equations is by graphing. This method is very impractical for finding the intersections, however, it does help to understand the solution obtained algebraically. Fortunately we have other methods for solving a system of equations such as elimination and substitution. Which method? Substitution or Elimination? Although elimination may seem easier than substitution, some systems must be solved using the substitution method. If one of the equations has both x an y squared and the other equation does not, use substitution. If both equations have the x and y squared, use elimination.

3. 5.5 Systems Involving Nonlinear Equations 3 Example 1. Solve the system: If one of the equations has a variable which is not squared use substitution. The reason we can not use elimination is we do not have like terms in the both equations. Let’s solve for x in the bottom equation. Now substitute -y - 2 for ‘x’ in the first equation. This will give an equation with only one variable. Then solve for y. Next, substitute both -3 and 1 for y in the equation x = -y - 2 to obtain the values for the variable x. The two solutions are the intersections of line and a circle. We will graph the equations to verify the answers are reasonable. Graphically, the solutions of (-3,1) and (1,-3) are correct. We also need to check the solutions by substituting them back into the original equations.

4. 5.5 Systems Involving Nonlinear Equations 4 Your Turn Problem #1 Answer: x y ● ● (2,3) (0,-1) Solve the system by substitution or elimination. Verify the solution by graphing both equations.

5. 5.5 Systems Involving Nonlinear Equations 5 Example 2. Solve the system: Since both of the equations has a variable which is not squared, use substitution. Substitute x-1 for ‘y’ in the first equation. This will give an equation with only one variable. Then solve for x. Next, substitute both values of x in one of the original equations to obtain the values for y. Obtaining a non real number solution means that the parabola and the line do not intersect in the real number plane. This quadratic equation is not factorable, use quadratic formula to solve. We can see the graphs do not intersect. However, we did obtain imaginary solutions.

6. 5.5 Systems Involving Nonlinear Equations 6 x y Answer: Note: Please do not feel that imaginary solutions have no meaning. For those involved in fields such as engineering and aeronautics, applications involving imaginary solutions are very important. Your Turn Problem #2 Solve the system by substitution or elimination. Verify the solution by graphing both equations.

7. 5.5 Systems Involving Nonlinear Equations 7 Example 3. Solve the system : Since the variables are all the same, elimination will work well. Multiply bottom row by -1 to eliminate the x when the two rows are added together. Substitute +1 and -1 for ‘y’ in either equation to find x.  Now, divide by -5 on both sides and use the square root property to solve for y. -5 The 1 st equation is a hyperbola: The center is (0,0), a =±2, b=±1. The 2 nd equation is a circle. The center is (0,0), r=3. The graphs do intersect. It is consistent with the solution set obtained using the elimination method.

8. 5.5 Systems Involving Nonlinear Equations 8 x y Answer ● ● ● ● The End B.R. 2-1-07 Your Turn Problem #3 Solve the system by substitution or elimination. Verify the solution by graphing both equations.