DAB - Feb. 20111 Solving Systems of Equations via Elimination D. Byrd February 2011.
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Transcript of DAB - Feb. 20111 Solving Systems of Equations via Elimination D. Byrd February 2011.
DAB - Feb. 2011 1
Solving Systems of Equations via Elimination
D. ByrdFebruary 2011
DAB - Feb. 2011 2
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 if1. two equations are interchanged2. an equation is multiplied/divided by a nonzero
constant3. one equation is added to another
– Rules 2 and 3 are often combined• “Add 3 times equation (b) to equation (a)”
DAB - Feb. 2011 3
Theorem on Equivalent Systems
• Do rules of Theorem on Equivalent Systems make sense?
• “An equivalent system results if…1. “two equations are interchanged”:
obvious!2. “an equation is multiplied (or divided)
by a nonzero constant”: pretty obvious3. “one equation is added to another”:
huh?– Demo with Geometers Sketchpad
DAB - Feb. 2011 4
Solving Systems by Elimination
• Example 1: elimination two different waysx + 3y = –12x – y = 5
DAB - Feb. 2011 5
Solving Systems by Elimination
• Example 23x + y = 66x + 2y = 12
• Example 33x + y = 66x + 2y = 20
DAB - Feb. 2011 6
Characteristics of Systems of Two Linear Equations in Two Unknowns
Ex. Graph No. of solutions
Classification
1 Intersecting lines
One Consistent
2 Identical lines Infinite number
Dependent & consistent
3 Parallel lines None Inconsistent
DAB - Feb. 2011 7
Word Problems #*(#$*@*&
DAB - Feb. 2011 8
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)