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Precalculus With Limits PPT that comes from the book and designed for teaching.
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- SUBSTITUTION- ELIMINATION- PARTIAL FRACTIONS- INEQUALITIES
CHAPTER 7 - REVIEW
Substitution
Example
Solve the system by the method of substitution.
52
4
yx
yx (1)
(2)
Solve (1) for x
4yx
Substitute this result into (2)
52)4( yy
543 y
3y
Back substitute to solve for x
4yx 1
Check solutions
5)3(21
431
)3 ,1(
Example
Solve the system by the method of substitution.
03
022yx
yx (1)
(2)
Solve (1) for x
yx 2
Substitute this result into (2)
0)2(3 2 yy
0)6( yy
0y
Back substitute to solve for x
OR 6y
yx 2
yx 2
0
12
Check solutions
000
000
06)12(3
0)6(2122
)6 ,12(
)0 ,0(
Example
-10 -5 0 5 10
-10
-5
0
5
10Solve the system graphically
2
12
yx
yx
Solve each equation for y
2/)1( xy
2xy
(1)
(2)
(1)(1)
(2) (2)
Graph each and find the intersection
(5, 3)
0 2 4 6 8 10 12 14 16 18 20
-10
-5
0
5
10
Example
Solve the system graphically
2
1
2
101142
yx
xy (1)
(2)
Solve each equation for y
)114( xy
2/)1( xy
(1)
(2)
Graph each and find the intersection
(1)
(2)
(15, 7)
(3, 1)
Example
A small fast food restaurant invests $5000 to produce a new fooditem that will sell for $3.49. Each item can be produced for $2.16.How many items must be sold to break even?
x16.2$5000$Cost
Let x be the number of items produced
x49.3$Revenue
Cost - RevenueProfit
)16.25000(49.3)( xxxP
Set profit equal to 0
0)16.25000(49.3 xx
500033.1 x
units 3760x
0 1000 2000 3000 4000 5000 6000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
P(x)
Break even point
units 3760x
Method of Elimination
Example
Solve the system by the method of elimination.
1352
253
yx
yx (1)
(2)
Add equations (1) & (2)
253 yx
1352 yx
15 5 x
3x
Back substitute to solve for y
25)3(3 y
5
7y
Check solutions
135
75)3(2
25
75)3(3
5
7 ,3
Example
Solve the system by the method of elimination.
1053
127
yx
yx (1)
(2)
Multiple (1) by -3
36213 yx
10 53 yx+
26 26 y
1y
Back substitute to solve for x
10 )1(53 x
5 x
Check solutions
10)1(5)5(3
12)1(75
)1 ,5(
Example
Solve the system by the method of elimination.
952
4113
yx
yx (1)
(2)
Multiple (1) by 2 and (2) by 3
8226 yx
27156 yx+
35 7 y
5 y
Back substitute to solve for x
4 )5(113 x
17 x
Check solutions
9)5(5)17(2
4)5(11)17(3
)5 ,17(
Interpretation of solutions
A linear system of equations is called consistent if it has at least one solution.
A consistent system with exactly one solution is independent. If it has infinitely many solutions it is dependent.
A system is called inconsistent if it has no solutions
Example
Two planes start from LA International airport and fly in opposite directions.The second plane leaves 30 minutes after the first plane, but its speed is
80 kilometers per hour faster than the first plane. Find the airspeed of each Plane if 2 hours after the first plane departs the planes are 32oo km apart.
Let v1 be the speed of the first plane and v2 be the speed of the second one.
km 3200hours) 1.5(hours) 2(distance 21 vv
8012 vv
km 3200)80(5.12 11 vv
30803.5 1 v
km/h 8801 v
8012 vv
km/h 9602 v
Example
A total of $32,000 is invested in two municipal bonds that pay 5.75%and 6.25% simple interest. The investor wants an annual interest Income of $1900 from the investments. What amount should be
invested in the 5.75% bond?
Let x be the amount invested in the 5.75% bond.
What is the amount invested in the 6.25% bond? x000,32
1900)000,32(0625.00575.0 incomeInterest xx
100005.0 x 000,20x
$20,000 should be invested in the 5.75% bond and 12,000 in the 6.25% one
Example
Use any method to solve the following system
2
1637
yx
yx (1)
(2)
Solve (2) for x
2yx
Substitute this result into (1)
163)2(7 yy
3010 y
3y
Back substitute to solve for x
2yx 1
Check solutions
231
16)3(3)1(7
)3 ,1(
Example
Use any method to solve the following system
175
634
yx
yx (1)
(2)
Multiple (1) by 5 and (2) by 4
301520 yx
42820 yx+
26 13 y
2 y
Back substitute to solve for x
6 )2(34 x
3 x
Check solutions
1)2(7)3(5
6)2(3)3(4
)2 ,3(
Example
Use any method to solve the following system
52
43
2
2
1
yx
yx (1)
(2)
Multiple (1) by 6
24)2(2)1(3 yx
2323 yx (3)
Solve (2) for x
52 yx
Plug this result into (3)
232)52(3 yy
88 y 1y
Back substitute to solve for x
52 yx 7 x
(7, 1) satisfies both equations
Example
Solve the system of linear equations
1
232
142
zyx
zyx
zyx (1)
(2)
(3)
Add (1) and (3)
142 zyx1 zyx+
053 yx (4)
Multiply (1) by 3 and add to (2)
33126 zyx
232 zyx+
5107 yx (5)
Multiple (4) by -2 and add to (5)
0106 yx
5107 yx+
5x3y
3z
Example
Solve the system of linear equations
3411
742
3235
zyx
zyx
zyx (1)
(2)
(3)
Multiply (2) by 2 and add to (1)
3235 zyx14284 zyx+
1759 yx (4)
Multiply (2) by 4 and add to (3)
284168 zyx3411 zyx+
3159 yx (5)
Subtract (4) from (5)
3159 yx
1759 yx-
140
NO SOLUTION
Example
A basketball team scored 70 points in a game. The number of 2 point basketswas two more than the number of free throws, and the number of freethrows was one more than two times the number of three point baskets. What combination of scoring accounted for the teams’ 70 points?
Let x represent the number of 2 point baskets, y represent the number of three point baskets, and z represent the number of free throws.
7032 zyx
2zx
12 yz
(1)
(3)
(2)
Substitute (2) into (1)
703)2(2 zyz
6633 zy (4)
Substitute (3) into (4)
66)12(33 yy
639 y 7y
15z 17x
The team made 17 two point baskets, 15 free throws, and 7 three points baskets
Partial Fractions - Example
Write the partial fraction decomposition of the following rational expression.
xx 14
72
Step 1: If the power in the numerator is greater than or equal to that in the denominator - divide
Step 2: Completely factor the denominator
)14(
7
14
72
xxxx
Step 3: decompose the rational expression
14)14(
7
x
B
x
A
xx
Partial Fractions - Example
Write the partial fraction decomposition of the following rational expression.
xx 14
72
Step 4: multiple by the LCD
)14(14)14(
7
xxx
B
x
A
xxBxxA )14(7
Step 5: solve – here plug in x = 0 to solve for A, and x = 14 to solve for B
2
1 and
2
1 BA
xxxx 2
1
)14(2
1
14
72
Example
Write the partial fraction decomposition of the following rational expression.
12
2
x
x
Since the power in the numerator is equal to that in the denominator, the first step is to divide (use long division)
1
11
1 22
2
xx
x
Factor the denominator
)1)(1(
11
12
2
xxx
x
Decompose
)1()1()1)(1(
1
x
B
x
A
xx
Example
Write the partial fraction decomposition of the following rational expression.
12
2
x
x
Step 4: multiple by the LCD
)1)(1()1()1()1)(1(
1
xxx
B
x
A
xx)1()1(1 xBxA
Step 5: solve – here plug in x = -1 to solve for A, and x = 1 to solve for B
2
1 and
2
1 BA
)1(2
1
)1(2
11
12
2
xxx
x
Example – Repeated Linear Factors
Write the partial fraction decomposition of the following rational expression.
12
322
xx
x
Factor
22 )1(
32
12
32
x
x
xx
x
Decompose – here we have arepeated linear factor
22 )1()1()1(
32
x
B
x
A
x
x
Multiple by the LCD
222
)1()1()1()1(
32
xx
B
x
A
x
x
BxAx )1(32
Solve – plug in x = 1 to get B, thenany x value to get A (2 is easy)
1 and 2 BA
22 )1(
1
)1(
2
12
32
xxxx
x
Example – Quadratic Factor
Write the partial fraction decomposition of the following rational expression.
xxx
xx
23
2 1
Factor
)1(
12
2
xxx
xx
Decompose – here we have alinear factor & a quadratic factor
)1()1(
122
2
xx
CBx
x
A
xxx
xx
Multiple by the LCD
)1()1()1(
1 222
2
xxxxx
CBx
x
A
xxx
xx
CxBxxxAxx 222 )1(1
AxCAxBAxx )()(1 22
To solve for A plug in x = 0
1A
Coefficients for the x2 and x termsmust be equal
1 BA 2B
1CA 0C
xxx
x
xxx
xx 1
1
21223
2
Example
Write the partial fraction decomposition of the following rational expression.
43
122
23
xx
xxx
Divide – degree of numeratoris greater than denominator
43
361
43
1222
23
xx
xx
xx
xxx
Factor the fraction & Decompose
14)1)(4(
36
43
362
x
B
x
A
xx
x
xx
x
Multiple by the LCD
)1)(4(14)1)(4(
36
xx
x
B
x
A
xx
x
)4()1(36 xBxAx
Solve – plug in x = -4 to get A, thenx = 1 to get B
5
3 and
5
27 BA
)1(5
3
)4(5
271
43
122
23
xxx
xx
xxx
Example
Write the partial fraction decomposition of the following rational expression.
632
623
xxx
x
Factor
)2(3)2(
6
632
6223
xxx
x
xxx
x
)2)(3(
6
632
6223
xx
x
xxx
x
23)2)(3(
622
x
C
x
BAx
xx
x
)2)(3(23)2)(3(
6 222
xx
x
C
x
BAx
xx
x
)3()2)((6 2 xCxBAxx
Plug in x = -2 to solve for C
4C
Coefficients for the x2 and x termsmust be equal
124226 22 xBAxBxAxx
122)2()4(6 2 BxBAxAx
04 A 4A
12 BA 9B
2
4
3
94
632
6223
xx
x
xxx
x
-10 -5 0 5 10
-20
-15
-10
-5
0
5
10
Inequalities
Sketch the graph of the inequality1535 yx
First sketch the graph of 1535 yx
Since the inequality is greater thanuse a dashed line to sketch the
graph. If it was just greater than or equal to use a solid line
Test a point above and below the lineto determine where to shade the
solution.
-3 -1 1 3 50
4
8
Example
Sketch the graph of the inequality
9)4()1( 22 yx
First sketch the graph of
9)4()1( 22 yx
Since it is greater than or equal to use a solid line
Test a point inside and outsideof the circle to determine
where to shade the solution.
Example
Sketch the graph and label the vertices of the solution set of the following system of inequalities.
935
62
yx
yx
First graph the line x - 2y = -6
-10 -5 0 5 10
-2
0
2
4
6
8 62 yx
Test a point above and below the lineto determine where to shade the
solution.
Example
Sketch the graph and label the vertices of the solution set of the following system of inequalities.
935
62
yx
yx
First graph the line 5x - 3y = -9
Test a point above and below the lineto determine where to shade the
solution.
-10 -5 0 5 10
-40
-20
0
20
40
935 yx
Example
Sketch the graph and label the vertices of the solution set of the following system of inequalities.
935
62
yx
yx
-10 -5 0 5 10
-2
0
2
4
6
8 62 yx
-10 -5 0 5 10
-40
-20
0
20
40
935 yx
Example
-10 -5 0
-15
-10
-5
0
5
10 935 yx
62 yx
Example
Sketch the graph and label the vertices of the solution set of the following system of inequalities.
0
1
32
42
y
x
xy
xy
First find the intersection points between the parabola and the line
3242 xx
0722 xx
Use quad formula
83.1or 83.3 xx
Intersection points are:(3.83, 10.66) and (-1.83, -0.66)
-5 0 5
-10
0
10
20
1x
32 xy
42 xy
(3.83, 10.66)
Find the vertices ofthe solution set
Find the intersection of x = -1 and the line
(-1, 1)(-1, 1)
Obviously (-1, 0) isa one of the vertices
Find the positivex intercept of the
parabola
(2, 0)
(2, 0)
0
1
32
42
y
x
xy
xy
Example
A person plans to invest up to $20,000 in two different interest-bearing accounts. Each account is to contain at least $5000. Moreover, the amountin one account should be at least twice the amount in the other one. Findand graph a system of inequalities to describe the various amounts that
can be deposited in each account.
Let x represent the amount in one account and y the amount in the other.
20000 yx
xy 2
5000x
5000y
0 5000 10000 15000 200000
5000
10000
15000
20000
250005000x
5000y
xy 2
20000 yx
20000 yx
xy 2
5000x
5000y