Download Section 9.2 Systems of Equations in Two Variables

Section 9.2 Systems of
Equations in Two Variables
Consider two lines in a plane.
What can happen?
Lively classroom discussion
ensues.
Independent/Dependent
• Independent - lines have
different slopes.
• Dependent - lines have the
same slope.
Inconsistent/Consistent
• Inconsistent - lines do not
intersect.
• Consistent - lines intersect.
Inconsistent/Consistent?
Independent/Dependent?
Case I
Consistent
Independent
One solution
Are they
Perpendicular?
Inconsistent/Consistent?
Independent/Dependent?
Case II
Are they
Parallel?
Inconsistent
Dependent
No solutions
Inconsistent/Consistent?
Independent/Dependent?
Case III
Consistent
Dependent
Infinite number of solutions
Are they the
same line?
Consider the System
x y 5
x y 3
x
0
2
5
y
5
3
0
x
0
2
3
y
-3
-1
0
4, 1
Using the Addition Method
x y 5
x y 3
2x  8
2
2
x4
(4)  y  5
4
4
y 1
4, 1
x  3 y  7
(
5
)

3
y


7
3x  2 y  23

5

5
2 x  6 y  14
9 x  6 y  69  3 y  12
3 3
11x  55
y4
11
11
5, 4
x5
Solve.
Example 2
5 y  2  3x
2x  1  3 y
• Use the seven steps
Addition Method - Seven Steps
• Step 1 Write both equations of the system
in standard form
Ax + By = C
• Step 2 If necessary multiply one or both
equations by appropriate numbers so that
the coefficients of x or y are negatives of
each other.
Addition Method - Seven Steps
• Step 3 Add the two equations to get
one equation with only one
variable.
• Step 4 Solve the equation from
Step 3.
• Step 5 Substitute the solution from
Step 4 into either of the original
equations.
Addition Method - Seven Steps
• Step 6 Solve the resulting
equation from Step 5 for the
remaining variable.
• Step 7 Check the answer.
Using the Addition Method
2x  4 y  5
4 x  8 y  9
 4 x  8 y  10
0  19
Multiply
by -2
• What does it
mean?
6x  3y  9
 8 x  4 y  12
2x  y  3
 2 x  y  3
00
Addition
Method
Multiply
by 1/3
Multiply
by 1/4
• What does it
mean?
The Substitution Method
2 x  3 y  6
Replace
y  3x y5with
y  3x  5
y

3(3)

3x5-5
2 x  3( 3x  5 )  6
2x  9 x  15  6
7 x  15  6
15  15
7 x  21
x3
y  95
y4
(3, 4)
Once upon a time…
• …Turbo was
in a barnyard
that was full
of pigs and
chickens.
Once upon a time…
• Turbo counted all of the heads of the pigs
and chickens. The result was 30.
• Turbo counted all of the feet of the pigs
and chickens. The result was 84.
• How many pigs were there?
• How many chickens were there?
Once upon a time…
• Let p = # of pigs
• Let c = # chickens
• Counting Equation
p + c = 30
• Value Equation
4p + 2c = 84
Pigs and Chickens
-2
p + c = 30
4p + 2c = 84
-2p – 2c = -60
2p = 24
p = 12
c = 18
John has $1.70, all in dimes and
nickels. He has a total of 22 coins.
How many of each kind does he have?
• Let d = number of dimes
• Let n = number of nickels
• Counting equation
d + n = 22
• Value equation
.10d + .05n = 1.70
12 + n = 22
d + n = 22
.10d + .05n = 1.70
100
n = 10
-5
10d + 5n = 170
-5d + -5n = -110
12 Dimes and
10 nickels
5d = 60
d = 12
Check
.10(12) + .05(10) = 1.70
1.20 + .50 = 1.70
Tickets to a movie cost $5.00 for adults and $3.00 for
children. If tickets were bought for 50 people for a total of
$196 how many adult tickets were sold and how many
children tickets were sold?
• Let A = the number of Adult tickets sold.
• Let C = the number of children tickets sold.
• Counting Equation
A + C = 50
• Value Equation
5A + 3C = 196
-3
A + C = 50
5A + 3C = 196
-3A – 3C = -150
2A = 46
23 + C = 50
C = 27
23 Adults and
27 Children
A = 23
Check
5(23) + 3(27) = 196
115 + 81 = 196
How much 20% alcohol solution and 50% alcohol solution
must be mixed to get 12 gallons of 30% alcohol solution?
• Let x = amount of 20%
• Let y = amount of 50%
• Counting Equation
x + y = 12
• Value Equation
.20x + .50y = .30(12) = 3.6
-2
10
x + y = 12
.20x + .50y = 3.6
2x + 5y = 36
-2x – 2y = -24
x + (4) = 12
x=8
8 gallons of 20%
4 gallons of 50%
3y = 12
y=4
Check
.2(8) + .5(4) = 3.6
1.6 + 2 = 3.6
Homework 9.2
Page 661
5,7, 11-15 odd
25, 47 – 53 odd
63, 66