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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)