One Step Equations – Addition
++++
++++
––––
––––
–––––
–––––
–––––
––––
––––
–––––
–––––
–––––
–––––
–––
–––
–––––
–––––
–––––
–––––
x + 8 = –15
+
+
–8
–8
x = –23
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
✓
•
•
•
•
Rewrite the equation
Replace x with –23
Do the math
Are both sides equal?
One Step Equations – Subtraction
–
–––
–––
––
x – 7 = –2
Draw a vertical line
and horizontal line
To get x by itself.
+
+++
+ ++
+
+++
+++
++
+++
+7
+7
x = +5
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
✓
•
•
•
•
Rewrite the equation
Replace x with 5
Do the math
Are both sides equal?
One Step Equations w/ Fractions – Adding/Subtracting
1 ●3
4 ●3
3
12
a=
+
+
a=
3

12
a=
2 ●4

3 ●4
8

12
3

12
11

12
The Lazier Way:
1.
Multiply the denominators (bottoms).
* That’s your new denominator (bottom).
2. Go to Step 4 to find the new numerators (tops)
A. Draw a vertical & horizontal.
B. Covert fractions to a common denominator.
Find the LCM
1. List multiples of both denominators
* 4: 4, 8, 12, 16, 20, ...
* 3: 3, 6, 9, 12, 18, 24 …
2. The smallest number in both lists is ..
3. ...so, that’s your new denominator.
4. To find your new numerators (tops):
I.
Whatever you multiplied to get the
new Denominator
II. ... multiply the numerator (top) by
the same thing.
C. Isolate a . Get rid of
.
D. Add its opposite to both sides.
One Step Equations w/ Fractions – Adding/Subtracting
1.
2.
3.
13
k
12
or
1
1  k
12
4.

23
k
35
5.
6.
y
89
24
or
y3
y

97
21
y4
20
a
21
7.
or
17
24
3
a
8
8.
j
65
42
or
13
21
j 1
j
134
21
or
23
42
j  6
8
21
One Step Equations – Multiplication
–––
–––– ––––
––––– –– –
––––– –––
––
––
––
––
––
––
––
–– ––
––
––
––
––
––
Draw a vertical line
and horizontal line
7b = –28
To get b by itself.
7
1. What’s happening
to b ?
7
* It’s b times 7.
* The opposite of
b times 7 is
b divided by 7 , so
b = –4
2. Divide both sides
by 7.
•–4
✓
3. Check
•
•
•
•
Rewrite the equation
Replace b with –4
Do the math
Are both sides equal?
One Step Equations – Division
Draw a vertical line
and horizontal line
– –– –
–––––
To get a by itself.
1. What’s happening
to a ?
* It’s divided by 3.
3•
= –9 • 3
a = –27
–?27
✓
* The opposite of
a divided by 3 is
multiplied by 3, so
2. Multiply both sides
by 3.
3. Check
• Rewrite the
equation
• Replace a with –27
• Do the math
• Are both sides
One Step Equations w/ Fractions – Multiplying/Dividing
2
x  10
3
2
x  10 3
3
2
3
2
Draw a vertical & horizontal
To get x by itself.
* Look at x. What’s happening to it ?
30
=
2
* It’s x times
... so to get rid of
x times
, ...
1. You have to MULTIPLY by
x =
Check
• Rewrite the equation
• Replace x with 15
• Do the math
• Are both sides equal?
2
x  10
3
2
x  10
3
30
3
or
10
x = 15
or
✓
A reciprocal is a flipped fraction
... and, the reciprocal of +
is +
... so, MULTIPLY both sides by
=
40
1
x =
the RECIPROCAL
or
x = –40
2. Cancel the opposites.
3. Multiply the fractions.
3
2
3
2
One Step Equations w/ Fractions – Multiplying/Dividing
m

5
12
63
21
or  1  m
32
32
21
5
or1  m
16
16
m

36
1
or  1
35
36
f 
55
7
or  3  m
16
16
m
20
5
or 
324
81

d
22
27
35
m
88
150 75
5
or or 5
28 14
14
Two–Step Equations – Multiplication
+ 7
+++
++++
–––
– –– –
+7
1.
Look at the variable side, find the constant,
and get rid of it first.
a constant is a number
+++
++++
–
without a variable
– it’s the “naked number”
2. To get rid of ‒7, add the opposite (+7)
3x
=
3
6
+++
+++
3
x
=
+
+
2
3. Cancel the opposites...
… bring down the variable term …then add.
4. To get rid of the coefficient, 3 ……
a coefficient is the number in front of the variable
… DIVIDE both sides by 3
Two–Step Equations – Division
‒8
2
++
++
++
++
‒8
x = ‒18
2
x = ‒ 36
2
– – 1.
–– ––
–– ––
–– ––
–– ––
2.
––
––
–– – – –– – –
–– – – –– – – 3.
–– –– –– ––
–– –– –– ––
– –– – – –– – –
– –– – – –– – – 4.
– –– –– –– ––
– –– –– –– ––
––
––
––
––
Look at the variable side, find the
constant, and get rid of it first.
To get rid of 8, add the opposite (‒8)
Cancel the opposites...
… drop the variable term …then add.
To get rid of x divided by 2, …
… MULTIPLY both sides by 2
Two–Step Equations – Multiplication
6 = 16 – a
‒16
‒16
Remember,
‒10 =
‒ a = ‒1a – 1 a
– 12 – 12
–4 =
– 2x
–2
–2
2 =
So, stick a 1 in
front of the a.
x
– 1a
‒1
‒10 =
‒1
10 =
4
a
Two–Step Equations – Division
9 = ‒ y + 12
2
x
2
x
+14 +14
= 22
= 44
2
7
y + 12
9 = –7
‒12
‒ 12
-7 ‒3=
‒7
21 = y
If you have
a negative
sign just
sitting in
front of a
fraction,
move it
next to the
constant.
x = –3
‒ 3 = ‒27 + y
8
192 = y
1.
EXAMPLE Writing
2
and Solving a Two-Step Equation
Negative six, increased by the product of four and a number, is negative twenty-two.
Negative six
increased by
–6
+6
+
The number
is negative
four.
2.
the product of four and a number
4n
is negative twenty-two.
4n
=
–22
+6
–16
4
=
=
4
–4
n
=
Fifteen is twenty-six less than the quotient of a number and negative three.
Fifteen is twenty-six less than the quotient of a number and negative three.
15
=
+ 26
(–3) 41
=
–123
=
–
26n
–3 + 26
n_ (–3)
–3
n
The number is
negative one
hundred
seventeen.
Writing and Solving a Two-Step Equation
Your online music website charges a monthly fee of $8, plus $0.35 for every song you
download. If you paid $13.25 last month, how many songs did you download?
monthly fee + songs
8
=
TOTAL
+ 0.35x = 13.25
You
downloaded
fifteen songs
1. Read it again, and pick out the TOTAL.
Set a blank equation equal to 13.25
2. Now, figure out HOW you get to that
total.
3. Solve for x (songs).
x = 15
Moe, Larry, and Curley are equal partners in a lemonade stand. To calculate each person’s
earnings, they’ll take the total money made, divide it by three, then subtract $2 (for supplies).
If each stooge got $43, what was the total money made?
total money –
3
x
3
supplies
–
x = 135
2
=
TOTAL
= 43
The total
money made
was $135.
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).
Solving Equations by Combining
Like Terms
3x +12 – 4x = 20 Look: There are 2 variable terms …
–1x +12
= 20
– 12
–1x
– 12
= 8
–1
x
… so, COMBINE LIKE TERMS first.
Remember,
–1
=
–8
‒1x = ‒x
but, just leave
the 1 there.
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 …then add.
4. To get rid of the coefficient, ‒1 … …
… DIVIDE both sides by ‒1
Solving Equations by Combining
Solve the equation.
1.
–6 = 11w –5w
w=–1
2.
4p +10 + p = 25
p=3
3.
Like Terms
–8r – 2 + 7r = – 9
r=7
Solving Equations
by using
EXAMPLE
3
Distributive Property
6n –2(n +1) = 26 Use Distributive property
6n –2(n +1) = 26
“outer times first”, then
“outer times second”,
Combine
like
terms.
6n –2n –2 = 26
4n
4n
– 2 = 26
+2 +2
Add 2 to each side.
= 28
Solve.
n
= 7
Solving Equations by using Distributive
1.
2.
3.
3(x – 9) = – 39
x= –4
–63 = –7(8 – p)
p = –1
Property
25 = –3(2x + 1)
x
14
=
or – 4
3
2
3
Solving Equations Using Square Roots
2
2
x = 64
2
x =
64
= +
– 64
x =+
–8
ANSWER
c = 0.0121
Take the square
root of both
sides.
Remember, real
numbers have 2
roots.
c2 = 0.0121
c =  0.11
The square root of 121 is
11, so the square root of
0.0121 must be .11
Don’t believe me?
Check it.
Evaluate
square roots.
The solutions are
8 and –8.
When you find the
square root of a decimal
number, pretend there is
no decimal place.
2
x =
196
361
196
x =
361
2
x=
196
361
When you find the
square root of a fraction,
find the square root of
each part separately.
x=
14
19
Solving Equations
Using
SquareRoots
Roots
Solving
Equations
Square
GUIDED PRACTICE
33. t2 = 36
34. k2 = 121
ANSWER
ANSWER
_6
t =+
x =+
– 11
ANSWER
ANSWER
x
x =+
– 14
=
+
– 0.09
15
EXAMPLE 4
Solving Equations
Using
SquareRoots
Roots
Solving
Equations
Square
37. On an amusement park ride, riders stand against a circular wall that spins.
At a certain speed, the floor drops out and the force of the rotation keeps the
riders pinned to the wall.
The model s = 4.95 r
gives the speed needed to keep riders pinned to the
wall. In the model, s is the speed in meters per second and r is the radius of the
ride in meters. Find the speed necessary to keep riders pinned to the wall of a
ride that has a radius of 2.61 meters.
s = 4.95 r
Write equation for speed of the ride.
= 4.95 2.61
Substitute 2.61 for r.
4.95 (1.62) Approximate the square root using a calculator.
= 8.019
ANSWER
Multiply.
The speed should be about 8 meters per second.
Solving Equations Square Roots
(
2
) ()
b = 64
2
(
2
) ( )
144 = a
2
(
2
)( )
2
x = 2.89
2
( ) ( )
81
=y
169
2
Solving Equations with Variables
GUIDED PRACTICE
*(not taught in Math 7)
1.
55 + 3x = 8x
– 3x
– 3x
on Both Sides*
What’s the goal?
Get the variables on one side...
…and the constants on the other.
55 = 5x
…so, if you get rid of 3x on the
left, you’ll have it.
11 = x
Solve.
or
x = 11
Solving
SolvingEquations
Equationswith
withVariables
Variables on
on Both
Both Sides*
Sides
GUIDED PRACTICE
*(not taught in Math 7)
2.
9x = 12x – 9
x=3
3.
–15x + 120 = 15x
4=x
Solving
SolvingEquations
Equationswith
withVariables
Variables on
on Both
Both Sides*
Sides
GUIDED PRACTICE
*(not taught in Math 7)
4.
4a + 5 = a + 11 1. Get the variables on one side...
…and the constants on the other.
–a
…but, which side for each?
–a
...it doesn’t really matter.
3a + 5 =
–5
3a
=
a=2
+ 11
–5
6
Hint: Move the smaller variable to
the larger variable’s side.
Subtract 5 to isolate the variable.
Solve.
Solving
SolvingEquations
Equationswith
withVariables
Variables on
on Both
Both Sides*
Sides
*(not taught in Math 7)
118.
3n + 7 = 2n –1
119.
n = –8
120. 11
+ 3x – 7 = 6x + 5 – 3x
there are no solutions for x
–6c + 1 = –9c + 7
c=2
121.
6x + 5 – 2x = 4 + 4x + 1
all values of x are solutions
Variables
Variables on
on Both
Both Sides*
Sides
Solving
SolvingPRACTICE
Equations
Equationswith
with
GUIDED
*(not taught in Math 7)
122. 4(w – 9) = 7w + 18 123. 2(y + 4) = –3y – 7
w = –18
y = –3
Solving Multi-Step Equations
GUIDED PRACTICE
117. 4a + 5 = a + 11
-a
-a
3a + 5 =
-5
3a
+ 11
-5
=
a=2
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 variable to the
larger variable’s side.
Subtract 5 to isolate the variable.
6 Solve.
Solving Multi-Step Equations
118.
3n + 7 = 2n –1
119.
n = –8
120. 11
+ 3x – 7 = 6x + 5 – 3x
there are no
solutions for x
–6c + 1 = –9c + 7
c=2
121.
6x + 5 – 2x = 4 + 4x + 1
all values of
x are
solutions
Solving Multi-Step Equations
GUIDED PRACTICE
122. 4(w – 9) = 7w + 18 123. 2(y + 4) = –3y – 7
w = –18
y = –3
Writing and Solving Multi-Step Equations
124. You and a friend are buying snowboarding gear. You buy a pair of goggles
that costs $39.95 and 4 tubes of wax. Your friend buys a helmet that costs
$54.95 and 2 tubes of wax. You each spend the same amount. Write and solve
an equation to find the price of one tube of wax.
Let x represent the price of one tube of wax.
39.95 + 4x = 54.95 + 2x
39.95 + 2x = 54.95
2x = 15.00
2x =15.00
2
2
x = 7.50
Write an equation.
Subtract 2x from each side.
Subtract 39.95 from each side.
Divide each side by 2
Solve.
ANSWER
The price of one tube of wax is $7.50.
One Step Equations w/ Fractions – Adding/Subtracting
3●3
1 ●4 4
=
6 ●4 24
9
+ a = 

8 ●3 24
4
4


24
24
13
a=

24
Check:
=

✓
13
24
=
The Lazier Way:
1.
Multiply the denominators (bottoms).
* That’s your new denominator (bottom).
2. Go to Step 4 to find the new numerators (tops)
A. Draw a vertical & horizontal.
B. Covert fractions to a common denominator.
The Right Way:
1. List multiples of both denominators (bottom)
* 6: 6, 12, 18, 24, 30, 36, 42, 48 ...
* 8: 8, 16, 24, 32, 40, 48, ...
2. The smallest number in both lists is ..
3. ...so, that’s your new denominator (bottom).
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.
C. Isolate a . Get rid of
.
D. Add its opposite to both sides.
One Step Equations w/ Fractions – Adding/Subtracting
1
+ a
4
2
8

= 
3
12
=
1

4
1
3

= 
4
12
Draw a vertical & horizontal
To get x by itself.
1. Get rid of +
• How? Add the OPPOSITE
to both sides
2. Cancel opposites.
3. Add
a
3
12

8
12
11

12
=
✓

8
12
•NOTE: With fractions,
you must find a
common
denominator .
Check
• Rewrite the equation
11
• Replace a with  12
• Do the math
• Are both sides equal?