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)