7'4 Systems of Nonlinear Equations

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7.4Systems of Nonlinear Equations

• Solve a nonlinear system of equations. A
  non-linear system is one in which one or more of
  the equations has a graph that is not linear.

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Nonlinear Systems of Equations

• The graphs of the equations in a nonlinear system
  of equations can have no point of intersection or
  one or more points of intersect.
• The coordinates of each point of intersection
  represent a solution of the system of equations.
• When no point of intersection exists, the system
  of equations has no real-number solution.
• We can solve nonlinear systems of equations by
  using the substitution or elimination (addition)
  method.

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Example A

• Solve the following system of equations
• We use the substitution method. First, we solve
  equation (2) for y.

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Example A continued

• Next, we substitute y 2x ? 3 in equation (1)
  and solve for x

• Now, we substitute these numbers for x in
  equation (2) and solve for y.
• y 2x ? 3
• y 2(0) ? 3
• y ?3
• (0, ?3) and
• Check each.

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Example A continued

• Check (0, ?3)
• Check

• Visualizing the Solution

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Example B

• Solve the following system of equations
• xy 4
• 3x 2y ?10
• Solve xy 4 for y.

• Substitute into 3x 2y ?10.

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Example B continued

• Use the quadratic formula

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Example B continued

• Substitute values of x to find y. 3x 2y
  ?10
• x ?4/3 x ?2
• The solutions are (?4/3, ?3) and (?2, ?2).

• Visualizing the Solution

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Example C

• Solve the system of equations
• Solve by elimination. Multiply equation (1) by 2
  and add to eliminate the y2 term.

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Example C continued

• Substituting x ?1 in equation (2) gives us
• The possible solutions are (1, 3), (?1, 3), (?1,
  ?3) and (1, ?3).

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Example C continued

• All four pairs check, so they are the solutions.

• Visualizing the Solution

• (1, 3), (?1, 3), (?1, ?3)
• and (1, ?3).

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Solve

• Solve using addition (multiply 2nd equation by -9
  before adding together)
• Note this is a circle that sits inside an
  ellipse they share the 2 points of
  intersection.

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Decompose