Download 7.3 Solving Systems by Linear Combinations

Algebra
7.3 Solving Linear Systems by
Linear Combinations
This is the third and final way to
solve linear systems.
graphing
The other two are ____________
substitution
and ______________.
Steps
1)
2)
3)
4)
5)
Arrange the equations with like terms in
columns.
Multiply one or both equations by a number to
obtain coefficients that are opposites for one
variable.
Add the equations. One variable will be
eliminated. Solve for the other.
Substitute this number into either original
equation and solve for the other variable.
Check.
Solve
-2x + 2y = -8
2x + 6y = -16
8y = -24
y = -3
2x + 6y = -16
2x + 6(-3) = -16
2x – 18 = -16
2x = 2
x=1
Solution: (1, -3)
Check: -2(1) + 2(-3) = -8
2(1) + 6(-3) = -16
Solve
3x = -6y + 12
-x + 3y = 6
Rewrite the top:
3x + 6y = 12
[ -x + 3y = 6 ]3
-3x + 9y = 18
15y = 30
y=2
-x + 3y = 6
-x + 3(2) = 6
-x + 6 = 6
-x = 0
x=0
Solution: (0, 2)
Check: 3(0) = -6(2) + 12
-(0) + 3(2) = 6
Solve
12x + 20y = 24
[ 3x + 5y = 6 ]4
[ -4x + 2y = 5 ]3
-12x + 6y = 15
26y = 39
y = 39/26
y = 3/2
-4x + 2(3/2) = 5
-4x + 3 = 5
-4x = 2
x = -½
Answer: (-½, 3/2)
Check: 12(-½) + 20(3/2) = 24
-4(-½) + 2(3/2) = 5
You try! Solve.
2x + 8y = -2
[ 5x + 4y = 3 ]-2
-10x - 8y = -6
-8x
= -8
x=1
2(1) + 8y = -2
2 + 8y = -2
8y = -4
y = -½
Answer: (1, -½)
Check: 2(1) + 8(-½) = -2
5(1) + 4(-½) = 3
A boat traveled from 24 miles downstream in 4 hours. It
took the boat 12 hours to return upstream. Find the speed
of the boat in still water(B) and the speed of the current(C).
Speed in still water + current speed = speed downstream
Speed in still water – current speed = speed upstream
B + C = 6 mph
B – C = 2 mph
2B
= 8 mph
B = 4 mph
The boat goes 4 mph.
4 mph + C = 6 mph
C = 2 mph
The current goes 2 mph.
Speed downstream is 24 miles/4 hours = 6 mph
Speed upstream is 24 miles/12 hours = 2 mph
HW
P. 414-415 (#9-41 4X) (#45-48)