Ten Algebra Rules! Charlotte Mar Solving Equations Example #1 5 > 9x x -5

Divide by -5.

-1 < 9x

Divide 9.

-1/9 < x d - 14 = 5

Add 14.

d = 19 Example #9 m - 48 = 29

Ad 48.

m = 77 First Rule Sixth Rule Example #1 Combining Like Terms > You can only add and subtract

terms with the same variable,

and you can only multiply and

divide terms by constants and

other terms with the same variable. For example:

49 + 30x - 67y + 98x

You can only add the 30x

and the 90x together since

their variable, x, is the same. For the problems,

55 + 79x + 52y

91 + 15q - 65p

51 - 65x

65 + 51y

You can't combine

anything in this equation because no

variable is the same. For the problems:

41(54 + 62x)

98 x 25y

59x - 54x

You can combine all of these because they're multiplying by a constant. For the examples:

84x(83y - 25) - 2

98 + 84(54y)

25- 56y x 65y

You can combine everything except for the terms that you have to subtract and add. Second Rule What you do to one side, you have to do to the other. If you add, subtract, multiply, or divide something from one side of an equation, you must do the same thing to the other side of the equation. Example #1 49 + 98y = -27 +180y

subtract 98y and add 27 to both sides.

76 = 82y

Divide 82.

y = 38/41 Example #2 39 + 8n = 25 - 2n

Subtract 39 and add 2n.

10n = -14

Divide by 10.

n = -1.4 Example #3 Example #4 3x + 7xy + 9x

Combine like terms.

12x + 7xy 4 - 3(4 - 2) = x - 7

Distributive property and determine exponent.

16 - 12 + 6 = x - 7

Combine like terms and add 7.

x = 17 2 y = 85 - 4y

Add 4y.

5y = 85

Divide 5.

y = 17 Example #5 Example #6 Example #7 2f + 6 = 19

Subtract 6.

2f = 13

Divide by 2.

f = 6.5 Example #8 4(8 - s) = 5s + 12

Distributive property.

32 - 4s = 5s + 12

Subtract 32 and subtract 5s.

-9s = -20

Divide by -9.

s = 2-/9 Example #10 21 + q = -18

Add 21.

q = 3 Third Rule Dividing by a negative on both sides with an inequality. When you multiply or divide by a negative when you have an inequality, you switch the inequality. Example #2 -14h > 91

Divide -14.

h < -6.5 Example #3 -5n > 10

Divide -5.

n < -2 Example #4 -81x + 2 < -7

Subtract 2.

-81x < -9

Divide -81.

x > 9 Example #5 -n/5 > 10

Multiply by -5.

n < -2 Example #6 -x/9 81

Divide -9.

x -9 > < - - Example #7 -p > 7

Divide by -1.

p < -7 Example #8 Fourth Rule Any number can be multiplied and added in any order. There's no order in which you need to multiply or add. Fifth Rule Order of Operations Always solve order of operations using the PEMDAS strategy. (Parenthesis, Exponents, Multiply and Divide, Add and Subtract) -7y > 14

Divide -7.

y < -2 Example #9 x/-9 > 18

Divide by -9.

x < -2 Example #10 -2/5p < -14

Multiply by -5/2.

p > 35. Example #1 Example #2 Example #3 Example #4 Example #5 Example #6 Example #7 Example #8 Example #9 Example #10 6 + 3 = 3 + 6 6 x 3 = 3 x 6 8 + 7 + 6 =

7 + 6 + 8 =

6 + 8 + 7 9 x 6 x 3 =

6 x 3 x 9 =

3 x 9 x 6 y + x = x + y 5 + h = h + 5 (a + b) + c =

a + (c + b) r + 9h = 9h + r 3n - 18 > t =

t < -18 + 3n -48 + 52j - 65 > 35 =

35 < 52j - 48 -65 36- 4 x 6 = 192x

MD

(36 - 24)/192 = x

S

10/192 = x

D

19.2 = x Example #1 Example #2 Example #3 Example #4 10 - 5 x 2 + (36/6) -3

P

10 - 5 x 2 + 6 - 3

M

10 - 10 + 6 - 3

AS

3 21x = -5 + 3

PE

21x = -5 + 3

21x = -5 + 243

A

21x = 238

D

x = 11.33 (9 - 2 x 4/2) 5 6 x (4 + 3 )/5

PE

6 x 85/5

MD

6 x 17

102 4 Example #5 Example #6 Example #7 Example #8 Example #9 Example #10 (10 x 4) + (9 + 3)

P

40 + 12

A

52 - 8 + 44/11 - 2

D

-8 + 4 - 2

AS

-6 6(5 + 2(3 - 8) - 3)

P

6(5 + 2(-5) - 3)

6(5 -10 - 3)

6(-5)

-30 (5 - 3) + 18

PE

2 + 324

8 + 324

A

332 3 2 3 -5 = 4 - 2(a - 5)

M

-5 = 4 -2a + 10

A

2a = 4 + 5 + 10

D

a = (4 + 5 + 10)/2

A

a = 19/2 6 + 3 = -3n

D

(6 + 3)/-3

A

9/-3

D

-3 To solve algebraic expressions, you must isolate the variable. Example #2 Example #3 Example #4 Example #5 Example #6 Example #7 Example #8 Example #9 Example #10 x - 3 = 6

Add 3.

x = 9 8(9 + 5) = x

Distribute.

72 + 40 = x

Add.

112 = x 2y - 4 = 12

Add 4.

2y = 16

Divide by 2.

y = 8 x + 6 = 6 - 3

Subtract 6.

x = 6 - 6 - 3

Subtract.

x = -3 7(x - 3) = 5

Distribute.

7x -21 = 5

Add 21.

7x = 26

Divide 7.

x = 26/7 5 = m-5 _____ 4 Multiply by 4.

20 = m - 5

Add 5.

25 = m 4h + 5 = 11

Subtract 5.

4h = 6

Divide by 4.

h = 2/3 18 = 4a - 2

Add 2.

20 = 4a

Divide by 4.

5 = a 6n + 3 = 3

Subtract 3.

6n = 0

Divide by 6.

n = 0 3x - 5 = 7

Add 5.

3x = 12

Divide by 3.

x = 4 Seventh Rule Exponents When raising a variable to a power, you multiply the exponents. When you want to multiply variables, you add their exponents. Example #1 Example #2 Example #3 Example #4 Example #5 Example #6 Example #7 Example #8 Example #9 Example #10 (x + y + z)

x + y + z 3 4 6 18 12 3 6r (- 4r + 5)

24r + 30r 2 3 2 (9r - t + 46) = 1 0 120x x 6 5 x = 120x 11 x x x x x = x 2 3 4 5 x x x x 15 230x + 439x - 580x = 89x 3 3 3 3 x x x x x = x x x x x 5 5x (9x + 8x - 2) =

45x + 40x - 10x 4 3 6 7 10 4x x = 5x x 3 4 5x + 6x = 5x +6x 2 5 2 5 Eighth Rule Cross Multiplying To find if fractions are equal you can cross multiply the numerators with the opposite denominators. Example #1 Example #2 Example #3 Example #4 Example #5 Example #6 Example #7 Example #8 Example #9 Example #10 1 50

2 100 x --- --- Multiply the 1 and 100, then the 2 and 50. If they're the same, then the fractions are equal. 100 = 100 Equal fractions. 5 6

8 7 x --- --- 35 = 48 Unequal fractions. / 6 1

18 6 x --- --- 6 = 18 Unequal fractions. 21 4

121 54 x --- --- 648 = 484 Unequal fractions. / / 54 23

65 89 x --- --- 4,806 = 1,495 Unequal fractions. / 9 27

1 3 x --- --- 27 = 27 Equal fractions. 96 12

8 1 x --- --- 96 = 96 Equal fractions. 8 5

48 30 x --- --- 240 = 240 Equal fractions. 54 52

65 63 x --- --- 3,402 = 3,380 / Unequal fractions. 36 21

45 55 x --- --- 1, 980 = 945 Unequal fractions. Ninth Rule Fractions To add and subtract fractions, you must find a LCM for the denominator. Example #1 Example #2 Example #3 Example #4 Example #5 Example #6 Example #7 Example #8 Example #9 Example #10 5 2

10 5 + --- --- 5 4

10 10 + --- --- 9

10 --- 6 3

12 6 - --- --- 6 6

12 12 - --- --- 0 1 3

3 5 + --- --- 5 9

15 15 + --- --- 14

15 --- 8 9

6 3 16 36

12 12 - - --- --- --- --- - 5

3 --- 9 2

3 6 - --- --- 36 4

12 12 - --- --- 8

3 --- 8 6

4 8 - --- --- 16 6

8 8 - --- --- 5

4 --- 9 1

5 10 18 1

10 10 17

10 --- --- --- --- --- - - 2 6

8 2 2 24

8 8 13

4 + + --- --- --- --- --- 6 5

1 2 12 5

2 2 7

2 - - --- --- --- --- --- 7 2

5 1 - --- --- 7 10

5 5 - --- --- 3

5 - --- Tenth Rule Additive Inverse If a number by the opposite of itself is added to it, it will end up as a sum of 0. Example #1 Example #2 Example #3 Example #4 Example #5 Example #6 Example #7 Example #8 Example #9 Example #10 3 - 3 = 0 -6 + 6 = 0 9 - 9 = 0 a - a = 0 -b + b = 0 -xyz + xyz = 0 khs + khs = 0 p - 2 -p = -2 \ / = 0 -y + 87 + y -9 = 78 / / = 0 9 + (-9) = 0

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