1. One Step Equations – Addition
One Step Equations, p.1
One Step Equations – Addition
Draw a vertical line – – – – – x
+ 8
= –15 + + + + – – – – –
and horizontal line + + + + – – – – –
To get x by itself. + +
1. Get rid of + 8 – – – – –8 –8 – – – – – – – – – – – –
How? Add the
opposite
but, what you do
to one side ...
... you’ve got to
do to the other – – – – – x
= –23 – – – – – – – – – –
2. Cancel opposites. – – – – – – – –
3. Add
4. Check – – –
Rewrite the equation – – – – – – – – – – ✓
Replace x with –23 – – – – – – – – – –
Do the math
One Step Equations, p.1
Are both sides equal?

2. One Step Equations – Subtraction
One Step Equations, p.1
One Step Equations – Subtraction
Draw a vertical line – x
– 7
= –2 – – –
and horizontal line – – – – –
To get x by itself.
1. Get rid of – 7 (or –7) + +7 +7 + + + + + + +
How? Add the
opposite + + + + + +
but, what you do
to one side ...
... you’ve got to
do to the other x
= +5
2. Cancel opposites. + + + + +
3. Add
4. Check
Rewrite the equation ✓
Replace x with 5
Do the math
One Step Equations, p.1
Are both sides equal?

3. One Step Equations w/ Fractions – Adding/Subtracting
A. Draw a vertical & horizontal. ●4 4 ●3 9
+ a = =
B. Covert fractions to a common denominator. ●4 ●3
The Right Way:
1. List multiples of both denominators (bottom)
* 6: 6, 12, 18, 24, 30, 36, 42,
* 8: 8, 16, 24, 32, 40, 48, ...
2. The smallest number in both lists is ..
a =
so, that’s your new denominator (bottom).
4. To find your new numerators (tops):
Check: =
Whatever you multiplied to get the new
denominator (bottom)... ✓ =
... multiply the numerator (top) by the
same thing.
C. Isolate a . Get rid of
The Lazier Way:
Multiply the denominators (bottoms).
* That’s your new denominator (bottom).
D. Add its opposite to both sides.
2. Go to Step 4 to find the new numerators (tops)

4. One Step Equations w/ Fractions – Adding/Subtracting1. 2. 3. 4. 5. 6. 7. 8.

5. One Step Equations – Multiplication
Draw a vertical line
and horizontal line – – –
7b = –28 – – – – – – – – – – – – – – – –
To get b by itself. – – – – – – – –
1. What’s happening
to b ? 7 7 – – – –
* It’s b times 7. – – – – – – – – – –
* The opposite of b times 7 is
b divided by 7 , so – – – – – – – –
b = –4 – – – – – –
Divide both sides
by 7.
3. Check
Rewrite the equation
•–4 ✓
Replace b with –4
Do the math
Are both sides equal?

6. 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
* The opposite of
a divided by 3 is
multiplied by 3, so
2. Multiply both sides by 3.
a = –27 ✓
3. Check
–27 ?
Rewrite the equation
Replace a with –27
Do the math
Are both sides equal?

7. One Step Equations w/ Fractions – Multiplying/Dividing
Draw a vertical & horizontal
To get x by itself.
* Look at x. What’s happening to it ? 30 =
* It’s x times
... so to get rid of
x times , ... 2
1. You have to MULTIPLY by
the RECIPROCAL x = x = 15 or
A reciprocal is a flipped fraction
Check
Rewrite the equation
Replace x with 15
... and, the reciprocal of is +
Do the math ✓
Are both sides equal? 30 3
... so, MULTIPLY both sides by or 10 40 =
2. Cancel the opposites. 1 x = x =
–40
3. Multiply the fractions. or

8. One Step Equations w/ Fractions – Multiplying/Dividing

9. Two–Step Equations – Multiplication Two–Step Equations – Division
3x – 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 = 6
3. Cancel the opposites... + + + + + +
… bring down the variable term
…then add. 3 3
4. To get rid of the coefficient, 3 …… x = 2 +
a coefficient is the number in front of the variable +
… DIVIDE both sides by 3
Two–Step Equations – Division
8 + x = ‒10
Look at the variable side, find the constant, and get rid of it first. – – + + – – – – – – 2 + + – – – – – – + + – – – – – –
‒ ‒8 + + – – – – – –
2. To get rid of 8, add the opposite (‒8) – – – –
x = 2 2
‒18 2 – – – – – – – –
3. Cancel the opposites... – – – – – – – – – – – – – – – –
… drop the variable term
…then add. – – – – – – – –
x =
‒ 36 – – – – – – – – –
4. To get rid of x divided by 2, … – – – – – – – – – – – – – – – – – –
… MULTIPLY both sides by 2 – – – – – – – – –

10. Two–Step Equations – Multiplication Two–Step Equations – Division6
– 12
‒16
‒16 4 – = 2x
‒10 = – a
Remember,
‒ a = ‒1a
So, stick a 1 in front of the a. 1 –2 –2 x
‒10 = – a 1 2 = ‒1 ‒1 4
x = –3
10 = a
Two–Step Equations – Division
9 = ‒ y + 12 7
‒ 3 = ‒27 + y 8
If you have a negative sign just sitting in front of a fraction, move it next to the constant.
+14
9 = y x 2
= 22 2 –7 2
‒ ‒ 12 x
= 44 = ‒7 –7 ‒3
21 = y
192
= y

11. Two–Step Equations with Fractional Coefficients
Step 1: Get rid of the CONSTANT on the variable side 5 7
4 + n = –11 5 7
–11 = 4 – n
– 4
– 4
SUBTRACT 4
from both sides.
– 4
– 4
Step 2: Get rid of the COEFFICIENT on the variable side ( ) ( ) 7 5 ) 5 7
n = –15 5 7 ( 7 5 ( ) n 7 5 – 7 5
–15 = – – 1
MULTIPLY BY THE
RECIPROCAL, – 7 5
–105 5
105 5
n =
= n
Step 3: Cancel opposites, multiply, then simplify. or or
n =
–21
= n 21

12. – –6 4n = –22 +6 +6 4n = –16 4 = 4 n = –4 n 15 26 –3 + 26 + 26 (–3)
Writing and Solving a Two–Step Equation
EXAMPLE 2 1.
Negative six, increased by the product of four and a number, is negative twenty–two.
Negative six
increased by
the product of four and a number is
negative twenty–two. –6 4n =
–22 +
4n = –16
The number is negative four. =
n = –4 2.
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. n – 15 26 =
The number is negative one hundred seventeen. –3
(–3) n_
(–3) = –3
–123 n =

13. 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?
1. Read it again, and pick out the TOTAL.
monthly fee songs = TOTAL
Set a blank equation equal to 13.25
0.35x
= 13.25
2. Now, figure out HOW you get to that
total.
You downloaded fifteen songs
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 – supplies = TOTAL 3
1. Read it again, and pick out the TOTAL.
Set a blank equation equal to 43
= 43
x – 2 3
2. Now, figure out HOW you get to that
total.
The total money made was $135.
3. Solve for x (total money made).
x = 135

14. Solving Equations by Combining Like Terms
3x +12 – 4x = 20
Look: There are 2 variable terms …
… so, COMBINE LIKE TERMS first.
Remember,
‒1x = ‒x
but, just leave the 1 there.
–1x = 20
– – 12
Look at the variable side, find the constant,
and get rid of it first.
–1x = 8
2. To get rid of +12, add the opposite (‒12) –1 –1
3. Cancel the opposites …
… bring down the variable term
…then add.
4. To get rid of the coefficient, ‒1 … …
x = –8
… DIVIDE both sides by ‒1

15. Solving Equations by Combining Like Terms
Solve the equation. 3.
–8r – 2 + 7r = – 9
–6 = 11w –5w 1. 2.
4p p = 25
w = – 1
p = 3
r = 7

16. Solving Equations by using Distributive Property
EXAMPLE 3
6n –2(n +1) = 26
Use Distributive property
“outer times first”, then
6n –2(n +1) = 26
“outer times second”,
Combine like terms. 6n
–2n –2
= 26
4n – 2 = 26
+ 2
+ 2
Add 2 to each side.
4n = 28
Solve.
n = 7

17. Solving Equations by using Distributive Property1. 2. 3.
3(x – 9) = – 39
–63 = –7(8 – p)
25 = –3(2x + 1)
x = or – 4
x = – 4
p = –1

18. Solving Equations Using Square Roots The solutions are2 = x 64
c2 =
When you find the square root of a decimal number, pretend there is no decimal place.
Take the square root of both sides.
c2 = 2 = x 64
The square root of 121 is 11, so the square root of must be .11
c =
Remember, real numbers have 2 roots. = + – 64
Don’t believe me?
Check it. = x + – 8
Evaluate square roots.
When you find the square root of a fraction, find the square root of each part separately.
x2 =
ANSWER
The solutions are
8 and –8.
x2 =
x =
x =

19. Solving Equations Square Roots Solving Equations Using Square Roots
GUIDED PRACTICE
33. t =
34. k = 2
121 + _ t = 6
ANSWER = x + – 11
ANSWER = x + –
0.09
ANSWER = x + – 14 15
ANSWER

20. Solving Equations Using Square Roots Solving Equations Square Roots
EXAMPLE 4
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 = 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. r s =
4.95
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
Multiply.
ANSWER
The speed should be about 8 meters per second.

21. Solving Equations Square Roots
( )2 ( )2
( )2 ( )2
( )2( )2
( )2 ( )2
b = 64
144 = a
x = 2.89
= y

22. Solving Equations with Variables on Both Sides*
*(not taught in Math 7)
GUIDED PRACTICE
55 + 3x = 8x 1.
What’s the goal?
Get the variables on one side...
– 3x
– 3x
…and the constants on the other.
…so, if you get rid of 3x on the left, you’ll have it.
55 = 5x
11 = x
Solve. or
x = 11

23. Solving Equations with Variables on Both Sides*
*(not taught in Math 7)
Solving Equations with Variables on Both Sides*
*(not taught in Math 7)
Solving Equations with Variables on Both Sides
GUIDED PRACTICE
–15x = 15x 3.
9x = 12x – 9 2.
x = 3
4 = x

24. ...it doesn’t really matter. 3a + 5 = + 11
Solving Equations with Variables on Both Sides*
*(not taught in Math 7)
Solving Equations with Variables on Both Sides
Solving Equations with Variables on Both Sides*
*(not taught in Math 7)
GUIDED PRACTICE
1. Get the variables on one side... 4.
4a + 5 = a + 11
…and the constants on the other.
…but, which side for each?
–a –a
...it doesn’t really matter.
3a + 5 =
Hint: Move the smaller variable to the larger variable’s side.
– – 5
Subtract 5 to isolate the variable.
3a =
Solve.
a = 2

25. Solving Equations with Variables on Both Sides
*(not taught in Math 7)
Solving Equations with Variables on Both Sides*
*(not taught in Math 7)
3n + 7 = 2n –1
118.
119.
–6c + 1 = –9c + 7
n = –8
c = 2
120.
11 + 3x – 7 = 6x + 5 – 3x
x + 5 – 2x = 4 + 4x + 1
there are no solutions for x
all values of x are solutions

26. 122. 4(w – 9) = 7w + 18 123. 2(y + 4) = –3y – 7 w = –18 y = –3
Solving Equations with Variables on Both Sides
Solving Equations with Variables on Both Sides*
*(not taught in Math 7)
GUIDED PRACTICE
(w – 9) = 7w + 18
(y + 4) = –3y – 7
w = –18
y = –3

27. Solving Multi–Step Equations
GUIDED PRACTICE
Get the variables on one side...
…and the constants on the other.
…but, which side for each?
117.
4a + 5 = a + 11
It doesn’t really matter.
–a –a
Hint: Move the smaller variable to the larger variable’s side.
3a + 5 =
– – 5
Subtract 5 to isolate the variable.
3a =
Solve.
a = 2

28. Solving Multi–Step Equations
3n + 7 = 2n –1
118.
119.
–6c + 1 = –9c + 7
n = –8
c = 2
120.
11 + 3x – 7 = 6x + 5 – 3x
x + 5 – 2x = 4 + 4x + 1
there are no solutions for x
all values of x are solutions

29. Solving Multi–Step Equations
GUIDED PRACTICE
(w – 9) = 7w + 18
(y + 4) = –3y – 7
y = –3
w = –18

30. Writing and Solving Multi–Step Equations
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.
x = x
Write an equation.
x = 54.95
Subtract 2x from each side.
2x = 15.00
Subtract from each side. 2x 2
15.00 =
Divide each side by 2
x = 7.50
Solve.
ANSWER
The price of one tube of wax is $7.50.

34. One Step Equations w/ Fractions – Adding/Subtracting
Draw a vertical & horizontal
+ a = =
To get x by itself.
1. Get rid of +
How? Add the OPPOSITE
to both sides =
2. Cancel opposites.
3. Add
NOTE: With fractions,
you must find a
common
denominator . a =
Check
Rewrite the equation
Replace a with ✓
Do the math
Are both sides equal?