Lesson 4.2 Solving Algebraic Equations - Maestra Floresmscieloflores.weebly.com/uploads/3/8/4/2/38422295/reteach4.2.7th.… · MIF_Reteach C2_Ch04.indd 98 15/12/11 1:50 AM - ØgÊg:S

Example

Lesson 4.2 Solving Algebraic Equations

2x ! 3 " 5

2x ! 3 " 5 2x ! 3 # 3 " 5 # 3 2x " 8 2x $ 2 " 8 $ 2 x " 4 x " 4 gives the solution of the equation 2x ! 3 " 5.

Check: Substitute 4 for x into the original equation.

2x ! 3 " 2 % 4 ! 3 " 5

When x " 4 , the equation 2x ! 3 " 5 is true . So x " 4 gives the correct solution.

Add 3 to both sides.

Simplify. Divide both sides by 2.

Simplify.

1. 6 # 8x " 24

6 # 8x " 24

6 # 8x ! " 24 ! Subtract from both sides.

8 x " Simplify.

8x $ " $ Divide both sides by .

x " Simplify.

x " gives the solution of the equation 6 # 8x " 24.

Check: Substitute for x into the original equation.

6 # 8x " 6 # 8 % "

When x " , the equation 6 # 8x " 24 is . So x "

gives the solution.

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92 Chapter 4 Lesson 4.2

MIF_Reteach C2_Ch04.indd 92 15/12/11 1:50 AM

2. 4 ! 12x " 20 3. !5y ! 5 " 10

25

12

x ! " 2

Method 1 Solve by balancing the equation.

25 x #

12 " 2

25 x #

12 ! 1

2 " 2 ! 12

25 x "

32

52 $

25 x "

52 $

32

x " 154

Subtract 12 from both sides.

Simplify.

Multiply both sides by 52, which is the

reciprocal of the coefficient 25.

Simplify.

Example

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4. 13

x ! 16

" 12


13

x ! 16

" 12

13

x ! 16

# " 12 #

13

x "

$ 13

x⎛⎝⎜

⎞⎠⎟ " $

x "

Subtract from both sides.

Simplify.

Multiply both sides by .

Simplify.

Example

Method 2 Solve by multiplying the equation by the least common denominator (LCD).

25 x !

12 " 2

10 $ 25 x ! 1

2 " 10$2

10 $ 25 x ! 10 $

12

" 10$2

4 x ! 5 " 20 4 x ! 5 # 5 " 20 # 5 4 x " 15 4 x % 4 " 15 % 4 x " 154

x "

154 gives the solution of the equation 2

5 x !

12

" 2.

Check: Substitute 154

for x into the original equation.

25 x !

12 "

25 $

154 !

12

" 32 !

12

" 2

When x "

154 , the equation 2

5 x ! 1

2 " 2 is true . x " 15

4 gives the solution.

Multiply both sides by 10, the LCD of 25 and 1

2.

Use the distributive property.

Simplify.


Divide both sides by 4.

Simplify.

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5. 25

x ! 110

" 15 6. 1

8 # 2

3w " 3

4

Method 2 Solve by multiplying the equation by the least common denominator (LCD).

13

x ! 16

" 12

$ 13

16

x !⎛⎝⎜

⎞⎠⎟ " $ 1

2⎛⎝⎜

⎞⎠⎟

$ 13

x ! $ 16

" $ 12⎛⎝⎜

⎞⎠⎟

2x ! "

2x ! # " #

2x "

2x % " %

x "

x " gives the solution of the equation 13

x ! 16

" 12.

Check: Substitute for x into the original equation.

13

x ! 16

" 13

x $ ! 16

"

When x " , the equation 13

x ! 16

" 12 is . x " gives

the solution.

Multiply both sides by LCD .


Simplify.


Simplify.

Divide both sides by .

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Example

Solve the equation with variables on the same side.

x ! 1.3 " 0.3x # 2.6

x ! 1.3 " 0.3x # 2.6 1.3x ! 1.3 # 2.6 Group like terms.

1.3x ! 1.3 " 1.3 # 2.6 " 1.3 Add 1.3 to both sides.

1.3x # 3.9 Simplify.

1.3x $ 1.3 # 3.9 $ 1.3 Divide both sides by 1.3.

x # 3 Simplify.

Complete.

7. 0.4x ! 3 " 1.2x # 0.6

0.4x ! 3 " 1.2x # 0.6

! 3 # 0.6 Group like terms.

! 3 " # 0.6 " Add to both sides.

# Simplify.

# Divide both sides by .

x # Simplify.

Solve each equation with variables on the same side.

8. 0.3x ! 0.1 " 0.2x # 1.4 9. y ! 1 ! 0.6y # 2.4

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Remember to check your solution by substituting

! 43

for x into the original

equation.

3 ! 4x " 2x # 11

Method 1 Isolate the variable on the left side of the equation.

3 ! 4x " 2x # 11 3 ! 4x ! 2x " 2x # 11 ! 2x Subtract 2x from both sides.

3 ! 6x " 11 Simplify.

3 ! 6x ! 3 " 11 ! 3 Subtract 3 to both sides.

!6x " 8 Simplify.

!6 x!6 " 8

! 6 Divide both sides by !6.

x " !43 Simplify.

Method 2 Isolate the variable on the right side of the equation.

3 ! 4x " 2x # 11 3 ! 4x # 4x " 2x # 11 # 4x Add 4x to both sides.

3 " 11 # 6x Simplify.

3 ! 11 " 11 # 6x ! 11 Subtract 11 from both sides. !8 " 6x Simplify.

!86 " !6 x

6 Divide both sides by 6.

!43 " x Simplify.

Example

10. 10 ! 4x " 2x # 16

10 ! 4x ! " 2x # 16 ! Subtract 2x from both sides.

10 ! " 16 Simplify.

10 ! ! " 16 ! Subtract 10 from both sides.

" 6 Simplify.

" Divide both sides by .

x " Simplify.

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11. 3x ! 5 " x # 1 12. 10 # 2y " 4 ! 3y

1.7x ! 1.1 " 0.9x # 0.5

1.7x ! 1.1 " 0.9x # 0.5 Isolate the variable on the left side.

1.7x ! 1.1 ! 0.9x " 0.9x # 0.5 ! 0.9x Subtract 0.9x from both sides.

0.8x ! 1.1 " 0.5 Simplify.

0.8x ! 1.1 # 1.1 " 0.5 # 1.1 Add 1.1 to both sides.

0.8x " 1.6 Simplify.

0 8x0.8 " 1.6

0.8 Divide both sides by 0.8.

x " 2 Simplify.

13. 3.1y # 1.2 " 2.3y # 2.8

14. 1.2 ! 3.2p " 5.2p # 4

15. 4.2m ! 0.6 " !1 ! 1.8m

Example

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56

13

34

12

m m ! " #


56m ! 1

3 " 34 m # 12

56 m ! 13 ! 34 m " 34 m # 1

2 ! 34 m Subtract 56

13

34

12

m m ! " # from both sides.

1012 m ! 1

3 ! 912 m " 1

2 The LCD of 56

13

34

12

m m ! " # and 56

13

34

12

m m ! " # is 12. 56

13

34

12

m m ! " # "1012

m; 56

13

34

12

m m ! " # " 912

m .

1

12 m ! 13 " 1

2 Group like terms.

112 m ! 1

3 # 13 " 1

2 # 13 Add 1

3 to both sides.

112 m " 56 Simplify.

12 $ 112 m " 12 $ 56 Multiply both sides by the reciprocal of 1

12 , 12.

m " 10

Method 2 Solve by multiplying both sides of the equation by the LCD.

56 m ! 1

3 " 34 m # 12

12 $ 56 m ! 1

3 " 12 $ 34 m # 1

2 Multiply both sides by 12, the LCD of 56

13

34

12

m m ! " # , 13, 5

613

34

12

m m ! " # , and 12

.

12 $ 56 m ! 12 $ 13 " 12 $ 34 m # 12 $ 1

2 Use the distributive property.

10m ! 4 " 9m # 6 Simplify.

10m ! 4 ! 9m " 9m # 6 ! 9m Subtract 9m from both sides.

m ! 4 " 6 Simplify.

m ! 4 # 4 " 6 # 4 Add 4 to both sides.

m " 10 Simplify.

Example

Remember to check your solution by substituting 10 for m into the original equation.

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16. 27

314

12

1y y! " #

27

1y ! " ! 3

1412

y ! !

"1# ! " 12

"1 ! " 12

"1 # ! " 12

#

y !

$ ( y) ! $ ( )

y !

17. ! " # 1

1025

34

14

x x

$

110

25

x ! ⎛⎝

⎞⎠ ! $ 3

414

x ! ⎛⎝

⎞⎠

$ 1

10x !

$

25

!

$

34

x "

$

14

2x # ! " 5

2x # # ! " 5 #

#8 ! " 5

#8 # 5 ! " 5 #

#13 ! x

! x

! x

18. 34

12

38

1p p ! " # 19. 25

16

310

46

! " # y y

Subtract 3

14y from both sides.

The LCD of 27

and 3

14 is 14.

27

y ! .

Group like terms.


Simplify.

Multiply both sides by $

Simplify.

Multiply both sides by the

LCD, $


Simplify.

Subtract 2x from both sides.

Simplify.



Simplify.

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15 (x ! 2) " 2

Method 1 Use the distributive property to expand the expression.

15 (x ! 2) " 2

15

# x ! 15

# 2 " 2

15

x ! 25

" 2

15

x ! 25 $ 25

" 2 $ 25

15

x " 85

x " 8 Method 2 Use inverse operations.

15(x ! 2) " 2

5 # 15(x ! 2) " 5 # 2

x ! 2 " 10 x ! 2 $ 2 " 10 $ 2 x " 8

Example


Simplify.


Simplify.

Multiply both sides by 5.

Simplify. Express in simplest form.

Multiply both sides by 5.

Simplify.

Subtract 2 from both sides.Simplify.

20. 13 (2y ! 6) " 3

13 (2y ! 6) " 3

# 13 (2y ! 6) " # 3 Multiply both sides by .

2y ! " 9 Simplify.

2y ! $ 6 " 9 $ Subtract 6 from both sides.

y " Simplify.

" Divide both sides by .

y " Simplify.

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2.4 ! 1.2(n ! 2) " 6

2.4 ! 1.2 # n ! 1.2 # 2 " 6 1.2n ! 4.8 " 6 1.2n ! 4.8 $ 4.8 " 6 $ 4.8 1.2n " 1.2

1.2n1.2 " 1.2

1.2 n " 1


Simplify.

Subtract 4.8 from both sides.

Simplify.

Divide both sides by 1.2.

Simplify.

21. 12

(3x $ 4) " 7 22. 5(m ! 4) " 15

23. 2x ! 4(6 $ x) " 30

2x ! 4(6 $ x) " 30

2x ! 4 # $ 4 # " 30

2x ! $ " 30

! " 30

! $ " 30 $

x "

"

x "

24. 2y ! 3(y $ 2) " $1 25. 3(2p $ 1) $ 5p " 4


Simplify.

Group like terms.


Simplify.


Simplify.

Example

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Documents

Lesson 4.2 Solving Algebraic Equations - Maestra Floresmscieloflores.weebly.com/uploads/3/8/4/2/38422295/reteach4.2.7th.… · MIF_Reteach C2_Ch04.indd 98 15/12/11 1:50 AM - ØgÊg:S