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Solving Systems of Equations via Elimination. D. Byrd February 2011. Equivalent Systems. Systems of equations are equivalent if they have the same solutions Theorem on Equivalent Systems (p. 574) Given a system of equations, an equivalent system results if two equations are interchanged

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equivalent systems
Equivalent Systems
  • Systems of equations are equivalent if they have the same solutions
  • Theorem on Equivalent Systems (p. 574)
    • Given a system of equations, an equivalent system results if
      • two equations are interchanged
      • an equation is multiplied/divided by a nonzero constant
      • one equation is added to another
    • Rules 2 and 3 are often combined
      • “Add 3 times equation (b) to equation (a)”
theorem on equivalent systems
Theorem on Equivalent Systems
  • Do rules of Theorem on Equivalent Systems make sense?
  • “An equivalent system results if…
      • “two equations are interchanged”: obvious!
      • “an equation is multiplied (or divided) by a nonzero constant”: pretty obvious
      • “one equation is added to another”: huh?
    • Demo with Geometers Sketchpad
solving systems by elimination
Solving Systems by Elimination
  • Example 1: elimination two different ways

x + 3y = –1

2x – y = 5

solving systems by elimination1
Solving Systems by Elimination
  • Example 2

3x + y = 6

6x + 2y = 12

  • Example 3

3x + y = 6

6x + 2y = 20

an application boat vs current speed
An Application: Boat vs. Current Speed
  • Motorboat at full throttle went 4 mi. upstream in 15 min.
  • Return trip (with same current, full throttle) took 12 min.
  • How fast was the current? The boat?
  • Use d = rt (distance = rate * time)