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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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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 • 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 • 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 • Example 1: elimination two different ways x + 3y = –1 2x – y = 5
Solving Systems by Elimination • Example 2 3x + y = 6 6x + 2y = 12 • Example 3 3x + y = 6 6x + 2y = 20
Characteristics of Systems of Two Linear Equations in Two Unknowns
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)