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Unformatted text preview: 4.1 Solving sysTems of linear equaTions in Two variables Goal: Solve Two equaTions wiTh Two unknowns
Purpose: Find The breakeven poinT where cosT is equal To revenue Review: A soluTion of an equaTion in Two variables is an ordered
pair (x,y) ThaT makes The equaTion True. An equaTion in Two
variables has an infiniTe number of soluTions. A sysTem of Two equaTions in Two variables is a pair of equaTions
wiTh The same Two variables (Two equaTions wiTh Two unknowns.) A soluTion of This sysTem is an ordered pair (x,y) ThaT makes boTh
equaTions True. This is The poinT where The graphs inTersecT. HW problem #1:
Is (2, 1) a soluTion of The sysTem? x  y = 3 2X4y=8 Replace x wiTh 2 and y wiTh —1 in each equaTion: X ——\( : 3 2X— LlY 7‘8
1 +131 3 2m we 131%
id 13 4+4 1% 3 :3 “(E=8 \[63 Yes (ll—l) '15 Q $Dlu¥10h OJ; ‘H’lﬂ. Sx/slem You can esTirnaTe The soluTion of a sysTem of equaTions by
graphing each equaTion. If The graphs are The same, The equaTions are dependent (infiniTe
number of soluTions) If The graphs are differenT, The equaTions are independent If They cross, There is one soluTion (The poinT where The
graphs inTersecT) If They are parallel, There is no soluTion HW problem #7:
Solve The sysTem x + y : 1 by graphing
X ~ 2y = 4 Graph boTh equaTions, and find The poinT where They inTersecT: ><+\{ =l x_z\{=q The graphﬁ ‘MTUSQQ’T OLJY (ix—l) Then check To make sure This ordered pair makes boTh equaTions True: (7—1—4) x+y "l x~ZY =4
2+(—~\’i 244.;qu
=l 1+1 =4
we a :4 HW: p.225 #1 — 14 SubsTiTuTion meThod:
STep 1: solve one of The equaTions for one of iTs variables STep 2: subsTiTuTe The expression for The variable found in sTep 1
inTo The oTher equaTion STep 3: solve for The remaining variable STep 4: subsTiTuTe The value for The known variable inTo one of
The equaTions and solve for The oTher variable HW problem #15: x + y =10 Y ' If an equaTion has fracTions, mulTiply by The LCD To geT rid of
fracTions. HW problem # 19: (lx+3),:_l\>“l
2 4 4 Elimination (Addition) method: Step 1: Rewrite each equation in standard form, Ax + By = C Step 2: If necessary, multiply one or both equations by a number
so that the coefficient of one variable in one equation is the
opposite of its coefficient in the other equation. Step 3: Add the equations. This eliminates one of the variables. Step 4: Solve the resulting equation for the remaining variable. Step 5: Substitute the value for theknown variable into either
original equation and solve for the other variable.
. x 'L
HW problem # 23: 2x — 4y: oCD (x+ 2y = 53
2X + 3‘ “) ©
AAA aqycﬁ‘xOﬂS togelnxey: HW: p. 226 #23  34 ...
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 Spring '08
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