Download Ch 8.4

Ch 8.1- System of Linear Equations in two variables
Ch 8.4 – Linear Inequalities
Linear Systems
System of Linear equations with Two Variables
Example
x+y=2
x–y=3
By substitution or elimination you will find x and y values.
By Elimination
x+ y = 2
x–y=3
Add
2x = 5
x = 5 ( Divide by 2 both sides )
2
Substitute x = 5 in the first equation
2
x+y=2
5 +y=2
2
5
5
y = 2 – ( subtract both sides)
2
2
y=-
1
2
Solution (
By Substitution
y = 2 – x (subtract x from both sides
x-y=3
x – ( 2 – x) = 3
x–2+x=3
2x – 2 = 3
2x = 5 ( add 2 both sides)
x=5
2
y=2-
y=–
5
2
1
2
Check
5
2
1
,- )
2
x+y=2
1
5
+ ( -2) = 2
2
x–y=3
5 1 4
= =2
2 2 2
Ch 8.1
Systems of Linear Equations (Pg 639)
Average weight of a thrush : t
Average weight of a robin : r
(Weight of thrushes) + ( weight of robins) = total weight
Thus 3t + 6r = 48
5t + 2r = 32
This pair of equations is an example of a linear system of two equations
in two unknowns ( 2 x 2 linear system)
A solution to the system is an ordered pair of numbers (t, r) that
satisfies both equations in the system
20
5t +2r = 32
10
5 (4) + 2 (6) = 32, True
(4, 6)
3t + 6r = 48
5
3(4) + 6(6) = 48, True
Conclusion : Both equations are true, so average weight of a thrush is 4 ounces,
and the average weight of a robin is 6 ounces
Using Graphing Calculator (Pg 641)
Enter Y1 = (21.06 – 3x)/ -2.8
Y2 = (5.3 – 2x)/1.2
Enter
press zoom 6
Enter
Final Graph
then press 2nd , calc
Enter
Inconsistent and Dependent Systems ( pg 643)
Dependent system
Inconsistent system
( Infinitely many solutions) ( parallel lines and no solution)
Consistent and Independent system
Intersect in one point and
Exactly one solution
Solution 2 x 2 Linear Systems( Pg 645)
1. Dependent system. All the solutions of one equation are
also solutions to the second equation and hence are solutions
of the system. The graphs of the two equations are the same
line. A dependent system has infinitely many solutions.
2. Inconsistent system. The graphs of the equations are
parallel lines and hence do not intersect. An inconsistent
system has no solutions.
3. Consistent and independent system. The graphs of the two
lines intersect in exactly one point. The system has exactly one
solution
Pg 644
Example 4
Press Y1 = -X + 5
Y2 = -X + 1.5
Enter Graph
10
Inconsistent and parallel lines
and no solution
9.4
-9.4
- 10
Example
Enter equation
Press window and enter
Press graph and calc
Consistent, and one solution ( .6, .4 )
Step 1
Example 7, Pg 646
Fraction of a cup of oats needed:
x
Fraction of a cup of wheat needed: y
Step 2
Cups
Grams of
Protein
Per Cup
Grams of
protein
Oats
x
11
11x
Wheat
y
8.5
8.5y
Mixture
1
-
10
First equation x + y = 1
Second Equation 11x + 8.5y = 10
Solve the system of graphing using Graphing calculator
y=-x+1
y = (10 – 11x) /8.5
X min = 0 X max = 0.94
Ymin = 0 Y max = 1
Francine needs 0.6 cups of oats and 0.4 cups of wheat
Solutions of Systems by Algebraic Methods
By Elimination
2x + 3y = 8
3x – 4y = - 5
Multiply first equation by 3 and second equation by –2
6x + 9y = 24
-6x + 8y = 10
Add
17 y = 34
y = 2 ( Divide by 17 )
Substitute y = 2 in the first equation 2x + 6 = 8
2x = 8- 6 ( subtract 6 )
2x = 2
x = 1 ( Divide by 2 )
The ordered pair (1, 2)
Solve by Linear Combination( example)
2p + 8q = 4
3
9
3
P = 2+q
3
2
6p + 8q = 12 ( Multiply the first equation by 9)
2p = 12 + 3q
(the second equation by 6)
2p – 3q = 12 ( subtract 3q)
Standard form
6p + 8q = 12
2p – 3q = 12
6p + 8q = 12
-6p + 9Q = -36 (Multiply the second equation by –3)
Add the equations
17q = -24, q = -24/17
Substitute q
2p = 12 + 3q
2p = 12 + 3( -24/17)
2p = 12 – 72/17
2p = 132/17, P = 66/17 The solution p = 66/17, q = -24/17
Pg 653, Ex 8.1, No. 23
a)
Supply equation y = 50x
b) Demand equation y = 2100 – 20x
c) The graph of y = 2100 – 20x has y intercept (0, 2100) and x-intercept (105, 0)
d) Xmin = 0, Xmax = 120
f) Ymin = 0, Ymax = 2500
Press Y enter equations
Press window, enter values
press 2nd and table
press graph and trace
a) The equilibrium price occurs at the intersection point (30,1500) in the above graph
b) To verify
Y = 50(30) = 1500
Y = 2100 – 20(30) = 1500
Yasuo should sell the wheat at 30 cents per busheland produce 1500 bushels
No.28. There are 42 passengers on an airplane flight for which first –class fare was
$400 and tourist fare was $320. If the revenue for the flight totaled $14,400, how
many first –class and how many tourist passengers paid for the flight?
A) Write algebraic expressions to fill in the table?
Number of
tickets
Cost per
ticket
Revenue
First -Class
x
400
400x
Tourist
y
320
320y
Total
42
-
14,400
[0, 50, 1] by [0, 50, 1]
b) Write an equation about the number of tickets sold:
x+ y = 42
c) Write a second equation about the revenue from the tickets: 400x + 320y= 14,400
d) Graph both equations and solve the system ( 12 first class passenger and 30 tourist
passenger )
8.4 Linear Inequalities in Two Variables ( pg 682)
12000
x + y > 10000
10000
y > - x + 10000
8000
4000
x + y = 10000
4000
8000
12000
To graph an Inequality Using a Test point ( pg 685)
1.
2.
3.
a)
b)
4)
Graph the corresponding equation to obtain the boundary
line
Choose a test point that does not lie on the boundary line.
Substitute the coordinates of the test point into the
inequality.
If the resulting statement is true, shade the half-plane that
includes the test point.
If the resulting statement is false, shade the half –plane
that does not include the test point.
If the inequality is strict, make the boundary line is a
dashed line
Use a test point ( Ex 2, pg 684)
3x – 2y < 6
First graph the line 3x – 2y = 6
The intercepts are (2, 0) and (0, -3) ( set x= 0 , and y = 0 )
Next choose a test point. Since (0,0) does not lie on the line,
we choose it as test point
3(0) – 2(0) < 6 True So we shade the half plane that
contains the test point (0,0)
3x – 2y < 6
(0,0)
3x – 2y = 6
Ex 8.4, No 33, Pg 689
Graph each system of inequalities and find the coordinates of the vertices
x+y>3
2y < x + 8
2y + 3x < 24
x > 0, y > 0
First graph x + y = 3
2y = x + 8
2y + 3x = 24
( 0, 4) is the point of intersection of x= 0 and 2y = x + 8
( 4, 6) is the point of intersection of the lines 2y = x + 8
and 2y + 3x = 24
( 8, 0) is the point of intersection of the lines
2y + 3x = 24 and y = 0
(3, 0) is the point of intersection of
the lines y = 0 and x + y = 3
y
2y = x + 8
(4, 6)
6
5
4
3
(0, 4)
2y + 3x = 24
(0, 3)
2
1
x+y=3
0 1 2 3 4 5 6 7 8
(3, 0)
(8, 0)
Ex 8.4 No 37( Pg 690)
Let x represent the number of student tickets sold
y represent the number of faculty tickets sold
The information that student tickets cost $1, faculty tickets cost $2, and the
receipts must be atleast $250, can be stated in the inequality
x + 2y > 250
So positive no of Tickets are sold x> 0 and y> 0
The system of inequalities is x + 2y > 250, x > 0, y > 0
x + 2y = 250
2y = 250 – x
x
y = 125 – 2
150
100
y = 125 – x
50
2
0
4
8
12