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Unit 3 Lecture 25
Solving Equations with
Algebraic Fractions
Solving Equations with Fractions
Lecture 25
1
Objectives
• To solve an equation with algebraic
fractions
Lecture 25
2
3x 7 x
 
5 2 10
Lecture 25
Solve the equation
3
3x 7 x
 
5 2 10
Solve the equation
Multiply both sides by
10
Lecture 25
4
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
Lecture 25
Solve the equation
Multiply both sides by
10
5
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
Solve the equation
Multiply both sides by
10
Eliminate the
denominators
Lecture 25
6
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
2
5
(10) 3 x
5

(10) 7
2

(10) x
Solve the equation
Multiply both sides by
10
Eliminate the
denominators
10
Lecture 25
7
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
2
5
(10) 3 x
5

(10) 7
2
6x + 35

(10) x
Solve the equation
Multiply both sides by
10
Eliminate the
denominators
10
= x
Lecture 25
8
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
2
5
(10) 3 x
5

(10) 7
2
6x + 35

(10) x
Solve the equation
Multiply both sides by
10
Eliminate the
denominators
10
= x
Lecture 25
Add –6x to both sides
9
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
2
5
(10) 3 x
5

(10) 7
2

(10) x
Solve the equation
Multiply both sides by
10
Eliminate the
denominators
10
6x + 35 = x
35 = -5x
Lecture 25
Add –6x to both sides
10
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
2
5
(10) 3 x
5

(10) 7
2

(10) x
Solve the equation
Multiply both sides by
10
Eliminate the
denominators
10
6x + 35 = x
35 = -5x
Lecture 25
Add –6x to both sides
Divide both sides by -5
11
3x 7 x
 
5 2 10
(10)3 x (10)7 (10) x


5
2
10
2
5
(10) 3 x
5

(10) 7
2

(10) x
Solve the equation
Multiply both sides by
10
Eliminate the
denominators
10
6x + 35 = x
35 = -5x
-7 = x
Lecture 25
Add –6x to both sides
Divide both sides by -5
Solution
12
4 4
8
 
x 3 5x
Solve the equation
Lecture 25
13
4 4
8
 
x 3 5x
Solve the equation
Multiply both sides by
15x
Lecture 25
14
4 4
8
 
x 3 5x
Solve the equation
(15 x )4 (15 x )4 (15 x )8


x
3
5x
Lecture 25
Multiply both sides by
15x
15
4 4
8
 
x 3 5x
Solve the equation
(15 x )4 (15 x )4 (15 x )8 Multiply both sides by


15x
x
3
5x
Eliminate the
denominators
Lecture 25
16
4 4
8
 
x 3 5x
Solve the
equation
(15 x )4 (15 x )4 (15 x )8


x
3
5x
5
3
(15 x )4 ( 15 x )4 ( 15 x )8


x
3
5x
Lecture 25
Multiply both sides
by 15x
Eliminate the
denominators
17
4 4
8
 
x 3 5x
Solve the
equation
(15 x )4 (15 x )4 (15 x )8


x
3
5x
5
3
(15 x )4 ( 15 x )4 ( 15 x )8


x
3
5x
60 - 20x
Multiply both sides
by 15x
Eliminate the
denominators
= 24
Lecture 25
18
4 4
8
 
x 3 5x
(15 x )4 (15 x )4 (15 x )8


x
3
5x
5
3
(15 x )4 ( 15 x )4 ( 15 x )8


x
3
5x
60 - 20x
= 24
Lecture 25
Solve the
equation
Multiply both sides
by 15x
Eliminate the
denominators
Add –60 to both
sides
19
4 4
8
 
x 3 5x
Solve the equation
(15 x )4 (15 x )4 (15 x )8 Multiply both sides by


15x
x
3
5x
5
3
Eliminate the
x )8 denominators
(15 x )4 ( 15 x )4 ( 15


x
3
5x
60 - 20x = 24
-20x = -36
Lecture 25
Add –60 to both sides
20
4 4
8
 
x 3 5x
(15 x )4 (15 x )4 (15 x )8


x
3
5x
5
3
(15 x )4 ( 15 x )4 ( 15 x )8


x
3
5x
60 - 20x
= 24
-20x = -36
Lecture 25
Solve the
equation
Multiply both sides
by 15x
Eliminate the
denominators
Add –60 to both
sides
Divide both sides
by -20
21
4 4
8
 
x 3 5x
(15 x )4 (15 x )4 (15 x )8


x
3
5x
5
3
(15 x )4 ( 15 x )4 ( 15 x )8


x
3
5x
60 - 20x
= 24
-20x = -36
x = 1.8
Lecture 25
Solve the
equation
Multiply both sides
by 15x
Eliminate the
denominators
Add –60 to both
sides
Divide both sides
by -20
Solution
22
2x x  2

5
6
Lecture 25
Solve the equation
23
2x x  2

5
6
Solve the equation
Multiply both sides by
30
Lecture 25
24
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
Lecture 25
Solve the equation
Multiply both sides by
30
25
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
Lecture 25
Solve the equation
Multiply both sides by
30
26
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
Lecture 25
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
27
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
12x = 5(x+2)
Lecture 25
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
28
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
12x = 5(x+2)
Lecture 25
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
Distributive Property
29
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
12x = 5(x+2)
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
Distributive Property
12x = 5x+10
Lecture 25
30
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
12x = 5(x+2)
12x = 5x+10
Lecture 25
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
Distributive Property
Add –5x to both sides
31
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
12x = 5(x+2)
12x = 5x+10
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
Distributive Property
Add –5x to both sides
7x = 10
Lecture 25
32
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
12x = 5(x+2)
12x = 5x+10
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
Distributive Property
Add –5x to both sides
Divide both sides by 7
7x = 10
Lecture 25
33
2x x  2

5
6
(30)2 x (30)( x  2)

5
6
6
5
(30) 2x (30) ( x  2)

5
6
12x = 5(x+2)
Solve the equation
Multiply both sides by
30
Eliminate the
denominators
Distributive Property
12x = 5x+10
Add –5x to both sides
7x = 10
x = 1.43
Divide both sides by 7
Solution
Lecture 25
34
y 4 y 3 5


2
9
18
Lecture 25
Solve the equation
35
y 4 y 3 5


2
9
18
Solve the equation
Multiply both sides by
18
Lecture 25
36
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
18
Lecture 25
37
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
18
Eliminate the
denominators
Lecture 25
38
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
Lecture 25
39
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


30
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Lecture 25
40
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


30
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Lecture 25
Distributive Property
41
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


30
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Distributive Property
9y – 36 - 2y + 6 = 5
Lecture 25
42
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


30
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Distributive Property
9y – 36 - 2y + 6 = 5
Combine like terms
Lecture 25
43
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Distributive Property
9y – 36 - 2y + 6 = 5
Combine like terms
7y - 30 = 5
Lecture 25
44
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Distributive Property
9y – 36 - 2y + 6 = 5
Combine like terms
7y - 30 = 5
Lecture 25
Add 30 to both sides
45
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Distributive Property
9y – 36 - 2y + 6 = 5
Combine like terms
7y - 30 = 5
7y = 35
Lecture 25
Add 30 to both sides
46
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9 (y-4) – 2 (y-3) = 5
Distributive Property
9y – 36 - 2y + 6 = 5
Combine like terms
7y - 30 = 5
7y = 35
Lecture 25
Add 30 to both sides
Divide both sides by 7
47
y 4 y 3 5
Solve the equation


2
9
18
(18)( y  4) (18)( y  3) (18)5 Multiply both sides by


18
2
9
1
8
9
2
(18) ( y  4) (18) ( y  3) (18) 5 Eliminate the
denominators


2
9
18
9(y-4) – 2(y – 3) = 5
Distributive Property
9y – 36 – 2y + 6 = 5
Combine like terms
7y - 30 = 5
7y = 35
y=5
Lecture 25
Add 30 to both sides
Divide both sides by 7
Solution
48