one step equations – addition –15 = –15 x + 8 + 8 –8–8–8–8 –8–8–8–8...

One Step Equations – AdditionOne Step Equations – Addition

= –15–15xx + 8+ 8

––88 ––88

–– –– –– –– –––– –– –– –– –––– –– –– –– –––– –– –– –– –––– –– ––

++++++++++++++++ –– –– –– –– ––

–– –– –– –– –––– –– –– –– ––

–– –– –– –––– –– –– –– –– –– –– ––

–– –– –– ––

= –23–23xx

Draw a vertical line and horizontal line

To get x by itself.

1. Get rid of + 8

• How? Add the opposite• but, what you do to one side ... ... you’ve got to do to the other

2. Cancel opposites.3. Add

4. Check

–– –– –– –– –––– –– –– –– –––– –– –– –– –––– –– –– –– –––– –– ––

• Replace x with –23• Do the math• Are both sides equal?

• Rewrite the equation

✓

+ +

One Step Equations – SubtractionOne Step Equations – Subtraction

= –2–2xx – – 77

+7+7 +7+7

–– ––

= +5xx

Draw a vertical line and horizontal line

To get x by itself.

1. Get rid of – 7 (or

–7) • How? Add the opposite• but, what you do to one side ... ... you’ve got to do to the other

2. Cancel opposites.3. Add

4. Check

• Replace x with 5• Do the math• Are both sides equal?

• Rewrite the equation

✓

–– –– –––– –– ––

++

––

++++++++++

++

++++++++ ++

++++

++++++++++

One Step Equations – MultiplicationOne Step Equations – Multiplication

–– –– –– –– –––– –– –– –– –––– –– –– ––

–––– –– –––– ––

–– –– –– –––– –– ––

77bb = = –28–28Draw a vertical line

and horizontal line

To get b by itself.

1. What’s 1. What’s happening happening

to to b b ? ? * It’s b times 7.b times 7.

* The opposite of b times 7 is

b divided by 7 b divided by 7 , so

2.2. Divide both Divide both sidessides

by by 77..

77–– –––– ––

–– –––– ––

–– –––– ––

–– –––– ––

–– –––– ––

–– –––– ––

–– –––– ––

77

bb = = –4–4

3. Check3. Check

• Replace b with –4• Do the math• Are both sides equal?

• Rewrite the equation•–

4 ✓

One Step Equations – DivisionOne Step Equations – DivisionDraw a vertical line and horizontal line

To get a by itself.

= = –9–9

–––– –– –– –– ––

–––– ––

* The opposite of

a divided by 3 is

multiplied by 3multiplied by 3, , so

2. Multiply 2. Multiply both sides by 3.3.

3 •3 • • • 33

1. What’s happening

to a ? * It’s divided by divided by

3.3.

a = a = –27–27

3. Check3. Check

• Replace a with –27• Do the math• Are both sides equal?

• Rewrite the equation✓?––

2727

+ aa = 8

3

6

1 A. Draw a vertical & horizontal.

B. Covert fractions to a common denominator.=

The Right Way:

The Lazier Way:

One Step Equations w/ Fractions – Adding/Subtracting

1. List multiples of both denominators (bottom)* 66: 6, 12, 18, 24, 30, 36, 42,

48 ...* 88: 8, 16, 24, 32, 40, 48, ...

2. The smallest number in both lists is ..

3. ...so, that’s your new denominator (bottom).

24 24

4. To find your new numerators (tops):

I. Whatever you multiplied to get the new

denominator (bottom)...II. ... multiply the numerator (top) by

the same thing.

●4

●4 4●3

●3 9

1. Multiply the denominators (bottoms). * That’s your new denominator

(bottom).2. Go to Step 4 to find the new numerators (tops)

24

4

24

4

aa = = 24

13

Check:

==24

13

==

✓C. Isolate aa . Get rid of .

D. Add its opposite to both sides.

One Step Equations w/ Fractions – Adding/SubtractingOne Step Equations w/ Fractions – Adding/Subtracting

+ a = 4

1

3

2

4

1

=

=12

3

12

8

12

11a =

4

1

CheckCheck

• Replace a with • Do the math• Are both sides equal?

• Rewrite the equation

12

8

12

3

12

8

12

11✓

Draw a vertical & horizontal

To get x by itself.

1. Get rid of +

• How? Add the OPPOSITE to bothboth sides

2. Cancel opposites.

3. Add

•NOTE: With fractions, you must find a commoncommon denominator denominator .

1. 2. 3. 4.

5. 6. 7. 8.

k35

23k

or

k

12

11

12

13

a8

3a

21

20

24

173

24

89

y

or

y

21

134

21

97

y

or

y

42

231

42

65

j

or

j

21

86

21

134

j

or

j

One Step Equations w/ Fractions – Adding/SubtractingOne Step Equations w/ Fractions – Adding/Subtracting

One Step Equations w/ Fractions – Multiplying/DividingOne Step Equations w/ Fractions – Multiplying/Dividing

To get x by itself.

* Look at x. What’s happening to it ?

* It’s x times ... so to get rid of

x times , ...

Draw a vertical & horizontal23

10x

23

10x

A reciprocal is a flipped fraction

... and, the reciprocal of + is +

... so, MULTIPLY both sides by

2

3

2

3

2

3

2

3

1. You have to MULTIPLY by the

RECIPROCAL

2. Cancel the opposites.

3. Multiply the fractions.

=30

2

x = or 15x =

CheckCheck

• Replace x with 15 • Do the math• Are both sides equal?

• Rewrite the equation2

310x

2

310x

30 3

or 10 ✓= 40

1

x = or –40x =

12

5m

36

11

35

36 orm

27

22f

mor 32

211

32

63mor

16

73

16

55m

88

35

mor 16

51

16

21

81

5

324

20 orm

14

55

14

75

28

150orord

One Step Equations w/ Fractions – Multiplying/DividingOne Step Equations w/ Fractions – Multiplying/Dividing

Two–Step Equations – Multiplication1. Look at the variable side, find the

constant, and get rid of it first.

2. To get rid of ‒7, add the opposite (+7)

+ 7 +7

3. Cancel the opposites...

… bring down the variable term

…then add. 3x =

4. To get rid of the coefficient, 3 ……

… DIVIDE both sides by 3

3 3

x = 2

‒8 ‒8

x = 2

2 2

x =

1. Look at the variable side, find the constant, and get rid of it first.

2. To get rid of 8, add the opposite (‒8) 3. Cancel the opposites...

… drop the variable term…then add.

4. To get rid of x divided by 2, …

… MULTIPLY both sides by 2

a constant is a number without a variable – it’s the “naked

number”

a coefficient is the number in front of the variable

6

‒18

Two–Step Equations – Division

–––– –– –––– ––

––––

++++++++++++++

++++++++++++++

++ ++++++ ++ ++

++++

++++++++

++++++++

–– ––

––

–– –––– ––

–––– ––

–– –––– –––– –––– ––

–– –––– –––– –––– ––

–– ––

––

–– ––––

––––

–– –––– –––– –––– –– ––

–––– ––

––

–– ––––

––––

–– –––– –––– –––– –– ––

––

–– ––––

–– ––––––

–––– –––– –––– –––– –– ––

–––– ––

–– –– ––––

––––

–– –––– –––– –––– –– ––

––‒ 36

Two–Step Equations – Multiplication

+14+14

= 222 2

Two–Step Equations – Division

– 12

– 12

4– = – 2x –2 –2

x2 =

4

x = 442

x

= 16 – a6‒16‒16

‒10 = – aRemember,

‒ a = ‒1aSo, stick a 1 in front of the a.

1

‒10 = – a1

‒1 ‒1

10 = a

9 = ‒ y + 12

7

If you have a

negative sign just sitting in front of a fraction, move it next to

the constant.

9 = y + 12 ‒12 ‒ 12

‒3= ‒7-7

21 = y

x =

‒ 3 = ‒27 + y

8

= y

–3

192

–7

EXAMPLE 2 Negative six, increased by the product of four and a number, is negative twenty-two.

n = –4

Negative six

+

the product of four and a number

–6 4n =

Fifteen is twenty-six less than the quotient of a number and negative three.

Writing and Solving a Two-Step EquationWriting and Solving a Two-Step Equation1.

2.

increased by isnegative twenty-two.

–22+6 +6 4n = –

16 4 = 4

The number is negative

four.

Fifteen15

is

=

twenty-six

26less than

–the quotient of a number and negative three.n

–3 + 26 + 26 41 n_

–3(–3)(–3) =

=–123 n

The number is negative

one hundred

seventeen.

Writing and Solving a Two-Step EquationWriting and Solving a Two-Step EquationYour online music website charges a monthly fee of $8, plus $0.35 for every songsong you download. If you paid $13.25 last month, how many

songssongs did you download?1. Read it again, and pick out the TOTAL.Set a blank equation equal to

13.25 = 13.25

2. Now, figure out HOW you get to that

total.

monthly fee + songs = TOTAL

8 + 0.35x

3. Solve for x (songs).x (songs).

Moe, Larry, and Curley are equal partners in a lemonade stand. To calculate each person’s earnings, they’ll take the total money madetotal money made, divide it by three, then subtract $2 (for supplies). If each stooge got

$43, what was the total money madetotal money made?1. Read it again, and pick out the TOTAL.Set a blank equation equal to

43 2. Now, figure out HOW you get to

that total.3. Solve for x (total money made).x (total money made).

= 43

total money – supplies = TOTAL 3

x – 2 3

x = 15

You downloaded fifteen

songs

x = 135

The total money

made was $135.

Solving Equations by Solving Equations by Combining Like Combining Like TermsTerms3x +12 – 4x =

20Look: There are 2 variable terms …

… so, COMBINE LIKE TERMS first.–1x +12 =

20

Remember,

‒1x = ‒x but, just

leave the 1 there.

– 12 – 12

–1x = 8–1 –1

x =

1. Look at the variable side, find the constant, and get rid of it first.

2. To get rid of +12, add the opposite (‒12)

3. Cancel the opposites …… bring down the variable term

4. To get rid of the coefficient, ‒1 … …

… DIVIDE both sides by ‒1

…then add.

–8

w = – 1

–6 = 11w –5w1.

Solve the equation.

p = 3

2. 4p +10 + p = 25

r = 7

3. –8r – 2 + 7r = – 9

Solving Equations by Solving Equations by Combining Like Combining Like TermsTerms

EXAMPLE 3

6n –2(n +1) = 26

Use Distributive property

Combine like terms.

4n = 28Solve.

n = 7

Add 22 to each side.

6n –2(n +1) = 26

““outer times first”, outer times first”, then

––2n2n

““outer times second”, outer times second”,

––226n = 26

4n 4n – 2 = 26 + 2 + 2

Solving Equations by using Solving Equations by using Distributive Distributive PropertyProperty

2

3x = or – 4x = – 4

3(x – 9) = – 39 25 = –3(2x + 1)–63 = –7(8 – p)

p = –1

1. 3.2.

Solving Equations by using Solving Equations by using Distributive Distributive PropertyProperty

3

14

GUIDED PRACTICE

55 + 3x = 8x1. What’s the goal?

– 3x – 3x Get the variables on one side...…and the constants on the other.

…so, if you get rid of 3x3x on the left, you’ll have it.

55 = 5x

Solve.11 = x

or

x = 11

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)

GUIDED PRACTICE

9x = 12x – 92.

x = 3

–15x + 120 = 15x3.

4 = x

Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)

GUIDED PRACTICE

4. 4a + 5 = a + 11

a = 2

1. Get the variables on one side... …and the constants on the other.

…but, which side for each?

...it doesn’t really matter.

Hint: Move the smaller Hint: Move the smaller variable to the larger variable to the larger variable’s side.variable’s side.

–a –a

3a + 5 = + 11

– 5 – 5

Subtract 55 to isolate the variable.

3a = 6 Solve.

Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)

–6c + 1 = –9c + 7119.

c = 2

120.

n = –8

3n + 7 = 2n –1118.

11 + 3x – 7 = 6x + 5 – 3x

121. 6x + 5 – 2x = 4 + 4x + 1

there are no solutions for x all values of x are solutions

Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)

y = –3w = –18

GUIDED PRACTICE

122. 4(w – 9) = 7w + 18123. 2(y + 4) = –3y – 7

Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides

Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*

*(not taught in Math 7)*(not taught in Math 7)