Lecture Notes: Solving Systems of Non-Linear Equations

Methods for Solving Non-Linear Systems

• Substitution: Solving one equation for one variable and substituting this into the other
• Elimination: Attempting to add/subtract equations to eliminate a variable
• Creative techniques may be necessary depending on the system

Example A

Equations:

1. x^2 - y^2 = 21
2. x + y = 7

Method Used: Substitution

1. Solve 2nd equation for x: x = 7 - y
2. Substitute x into the 1st equation:
   • (7 - y)^2 - y^2 = 21
   • Expand and simplify:
     • 49 - 14y + y^2 - y^2 = 21
     • 49 - 14y = 21
   • Solve for y:
     • -14y = -28
     • y = 2
   • Solve for x using x = 7 - y:
     • x = 7 - 2 = 5
3. Solution Set: (5, 2)
   • Verify graphically: x^2 - y^2 = 21 (hyperbola) and x + y = 7 (line)

Example B

Equations:

1. 2y^2 + 3xy - 3xy + 6y + 2x + 4 = 0
2. 2x - 3y + 4 = 0

Method Used: Substitution

1. Solve 2nd equation for x: x = (3y - 4) / 2
2. Substitute x into the 1st equation:
   • 2y^2 - 3((3y - 4) / 2)y +6y + 2((3y - 4)/2) + 4 = 0
   • Simplify and distribute:
     • 2y^2 - 9/2y^2 + 6y + 6y + 3y - 4 = 0
   • Combine like terms:
     • -5/2y^2 + 15y = 0
   • Factor out y:
     • y(-5/2y + 15) = 0
   • Solve for y: y = 0 or y = 6
   • Substitute y back to find x:
     • y = 0 => x = -2
     • y = 6 => x = 7
3. Solution Set: (-2, 0) and (7, 6)

Example C

Equations:

1. y^2 - x^2 + 4 = 0
2. 2x^2 + 3y^2 = 6

Method Used: Elimination

1. Rewrite 1st equation:
   • -x^2 + y^2 = -4
2. Multiply 1st equation by 2 and add to the 2nd equation:
   • -2x^2 + 2y^2 = -8 and 2x^2 + 3y^2 = 6
   • Combine:
     • 5y^2 = -2 (Impossible)
3. Solution Set: No Real Solution

Example D

Equations:

1. 2/x^2 - 3/y^2 + 1 = 0
2. 6/x^2 - 7/y^2 + 2 = 0

Method Used: Elimination

1. Multiply the first equation by -3 and add to the second:
   • -6/x^2 + 9/y^2 - 3 = 0
   • 6/x^2 - 7/y^2 + 2 = 0
   • Combine:
     • 2/y^2 - 1 = 0
     • y = ±√2
2. Substitute y into the first equation to solve for x:
   • Solve for x: x = ±2
3. Solution Set: (2, √2), (-2, -√2), (2, -√2), and (-2, √2)

Example E

Equations:

1. x^2 - xy - 2y^2 = 0
2. xy + x + 6 = 0

Method Used: Factoring and Substitution

1. Factor 1st equation:
   • (x - 2y)(x + y) = 0
   • x = 2y or x = -y
2. Substitute each into 2nd equation:
   • x = 2y:
     • 2y^2 + 2y + 6 = 0 (No Real Solution)
   • x = -y:
     • -y^2 - y + 6 = 0
   • Solve:
     • y = -3, y = 2
     • x = 3, x = -2
3. Solution Set: (3, -3) and (-2, 2)

Summary

• Non-linear systems require a combination of substitution, elimination, and sometimes creative/clever techniques
• Verify solutions graphically wherever possible

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