LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

2.6 Nonlinear Systems of Equations Solving Nonlinear Systems with Real Solutions Solving Nonlinear Equations with Nonreal Complex Solutions

Solve the system. Solution SOLVING A NONLINEAR SYSTEMS BY SUBSTITUTION Example 1 Solve the system. (1) (2) Solution When one of the equations in a nonlinear system is linear, it is usually best to begin by solving the linear equation for one of the variables. Solve equation (2) for y. Substitute this result for y in equation (1).

SOLVING A NONLINEAR SYSTEMS BY SUBSTITUTION Example 1 (1) Let y = – 2 – x. Distributive property Standard form Factor. Zero factor property

SOLVING A NONLINEAR SYSTEMS BY SUBSTITUTION Example 1 Substituting – 2 for x in equation (2) gives y = 0. If x = 1, then y = – 3. The solution set of the given system is {(– 2, 0),(1, – 3)}.

Visualizing Graphs Visualizing the types of graphs involved in a nonlinear system helps predict the possible numbers of ordered pairs of real numbers that may be in the solution set of the system.

Solve the system. Solution SOLVING A NONLINEAR SYSTEM BY ELIMINATION Example 2 Solve the system. (1) (2) Solution The graph of equation (1) is a circle and, as we will learn in the future units, the graph of equation (2) is a hyperbola. These graphs may intersect in 0, 1, 2, 3, or 4 points. We add to eliminate y2.

Remember to find both square roots. SOLVING A NONLINEAR SYSTEM BY ELIMINATION Example 2 (1) (2) Add. Divide by 3. Remember to find both square roots. Square root property

Find y by substituting back into equation (1). SOLVING A NONLINEAR SYSTEM BY ELIMINATION Example 2 Find y by substituting back into equation (1). If x = 2, then If x = − 2, then The solutions are (2, 0) and (– 2, 0) so the solution set is {(2, 0), (– 2, 0)}.

Solve the system. Solution Example 3 SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS Example 3 Solve the system. (1) (2) Solution (1) Multiply (2) by – 1 Add. (3) Solve for y(x ≠ 0).

SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS Example 3 Now substitute for y in either equation (1) or (2). We use equation (2). Let y = in (2). Multiply and square. Multiply x2 to clear fractions.

Example 3 SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS This equation is in quadratic form. Subtract 6x2. Factor. Zero factor property. Square root property; For each equation, include both square roots.

SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS Example 3 Substitute these x-values into equation (4) to find corresponding values of y. If , then If , then If , then If , then

The solution set of the system is SOLVING A NONLINEAR SYSTEM BY A COMBINATION OF METHODS Example 3 The solution set of the system is If , then If , then If , then If , then

Solve the system. Solution Example 4 SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION Example 4 Solve the system. (1) (2) Solution Use the substitution method. Solving equation (2) for x gives (3) Since x  0 for all x, 4 – y  0 and thus y  4. In equation (1), the first term is x2, which is the same as x2.

Remember the middle term. SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION Example 4 Therefore, Substitute 4 – y in (1). Square the binomial. Combine terms. Remember the middle term. Factor. Zero factor property.

SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION Example 4 To solve for the corresponding values of x, use either equation (1) or (2). We use equation (1). If y = 0, then If y = 4, then

SOLVING A NONLINEAR SYSTEM WITH AN ABSOLUTE VALUE EQUATION Example 4 The solution set, {(4,0), (– 4, 0), (0, 4)}, includes the points of intersection shown in the graph.

Solve the system. Solution Example 5 SOLVING A NONLINEAR SYSTEM WITH NONREAL COMPLEX NUMBERS IN ITS SOLUTIONS Example 5 Solve the system. (1) (2) Solution Multiply (1) by – 3 . (2) Add. Square root property

To find the corresponding values of y, substitute into equation (1). SOLVING A NONLINEAR SYSTEM WITH NONREAL COMPLEX NUMBERS IN ITS SOLUTIONS Example 5 To find the corresponding values of y, substitute into equation (1). If x = 2i, then If x = – 2i, then i2 = – 4

SOLVING A NONLINEAR SYSTEM WITH NONREAL COMPLEX NUMBERS IN ITS SOLUTIONS Example 5 Checking the solutions in the given system shows that the solution set is Note that these solutions with nonreal complex number components do not appear as intersection points on the graph of the system.