Algebra Basics – One-Step Equations
In algebra, one-step equations can often be solved “in your head,” without
using formal algebraic steps. However, it’s important to learn how to use
formal algebraic steps with these simpler equations so if the numbers get
tricky to do in your head, you’ll have a backup plan. Also, as equations
get more complex (with more steps), it’s less likely you’ll be able to solve
them in your head. So, it’s important to know how to use algebraic steps
to solve them.
Since the variable in an equation represents what we don’t know (and also what we’re trying to
find!), it’s important to know the main goal, which is…
GOAL: Get the variable by itself on its own side of the equation.
If we accomplish this goal, then the numerical solution will be on the other side of the equation.
How we accomplish this goal varies from problem to problem, but the main goal remains the same.
Do your best to remember it always and you’ll be a better equation solver.
Some Questions and Answers
Question: Why is the goal to get the variable by itself?
Answer: If you get the variable by itself on one side, then a number is on the other side. This
means you now know that the variable is equal to that number!
Question: So, in general, how do we get the variable by itself?
Answer: We focus on what we want to “get rid of” that is near the variable and make this happen
by doing the opposite operation that we see on both sides of the equation.
Question: Why do we have to do the opposite operation on both sides of the equation if we only
want to get rid of something on one side?
Answer: An equation is like a balance scale or a see-saw. To keep it balanced, whatever we do to
one side of the equation, we must also do to the other side to keep the equation balanced.
(Otherwise, the equation makes no mathematical sense and is useless to us.)
Over (for some examples)
Important Note: The equals sign is always the border
between the left and right sides of an equation.
=
Example 1: If the equation that we’re trying to solve is
15 36x
, first we focus on the variable
x
.
It’s a great idea to circle it. We want to get it by itself on the left; it will then be equal to whatever
number is on the right. It’s important to remember, though, that we must use correct mathematics
(algebra, really) to make this happen.
So, we have
15 36x
. We want to get rid of the
15
so we do the opposite, which is to subtract
15 from the left side of the equation ( Remember that the equals sign is always the border
between the left and right sides of an equation.) However, an equation is like a balance scale or a
see-saw. To keep it balanced, we must also subtract 15 from the right side. The work usually
looks like this:
15 36
15 15
21
x
x


Check: It’s a great idea to check to make sure your solution actually works. Do this by replacing
x
in the original equation with the number you got for your solution (in this case, 21) and then see if
the equation is true…
15 36
(21) 15 36
36 36
x 

Good Job!
Example 2:
Example 3:
43
33
7
7
x
x
x



Example 4:
3.2 8
3.2 3.2
11.2
x
x

Name:_______________________________
Algebra Basics – One-Step Equations (+/-) (A)
Read and follow these directions for EACH exercise:
1. Show appropriate adding or subtracting on both sides of the equation.
2. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
5 10x
2.
4.7 11x
3.
2.5 22x
4.
5.
55 98x
6.
9 16x
7.
22 33x
8.
22 33x
9.
14 9x
10.
59x
11.
10 19x
12.
10 35x
13.
13 49x
14.
15 2x
15.
60 30x
Name:_______________________________
Algebra Basics – One-Step Equations (+/-) (B)
Read and follow these directions for EACH exercise:
1. Show appropriate adding or subtracting on both sides of the equation.
2. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
7 19x
2.
5.1 15x
3.
4.5 18x
4.
32 27x
5.
76 82x
6.
4 13x
7.
26 51x
8.
23 36x
9.
34 19x
10.
8 12x
11.
5 17x 
12.
25 11x
13.
14 13x
14.
16 2x
15.
08x
Name:_______________________________
Algebra Basics – One-Step Equations (x/) (C)
Read and follow these directions for EACH exercise:
1. Show appropriate multiplication or division on both sides of the equation.
2. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
4 44x
2.
4 43x
3.
2.5 20x
4.
93x
5.
5 65x
6.
7 16x
7.
342x
8.
70x
9.
10.
44
2
x
11.
5
5
x
12.
20
3
x

13.
9
2
x

14.
4.5
7
x
15.
0.5
4
x
16.
44
11
x
17.
0
12
x
18.
2
43
x

Name:_______________________________
Algebra Basics – One-Step Equations (x/) (D)
Read and follow these directions for EACH exercise:
1. Show appropriate multiplication or division on both sides of the equation.
2. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
2 24x
2.
2 13x
3.
1.5 9x
4.
18 3x
5.
5 75x
6.
3 14x
7.
24x
8.
90x
9.
10.
14
3
x
11.
7
6
x
12.
10
4
x

13.
11
3
x

14.
9.5
2
x
15.
1.5
7
x
16.
9
12
x
17.
2
32
x

18.
0
13
x
Name:_______________________________
Algebra Basics – One-Step Equations (+/-/x/) (E)
Read and follow these directions for EACH exercise:
1. Show appropriate addition, subtraction, multiplication or division on both sides of the
equation.
2. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
73y
2.
46h
3.
8 24x
4.
5
2
x
5.
65t
6.
4
9
n
7.
128 4x
8.
82c
9.
85k
10.
8
3
x

11.
4.5 9y 
12.
12 2.5y
13.
0
9
x
14.
44w
15.
88x
16.
12 0w
17.
1.5 6c
18.
6
1.5
x

Name:_______________________________
Algebra Basics – One-Step Equations (+/-/x/) (F)
Read and follow these directions for EACH exercise:
1. Show appropriate addition, subtraction, multiplication or division on both sides of the
equation.
2. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
2.
4 16h
3.
6 30x
4.
6
3
x
5.
86t
6.
7
4
n
7.
65 5x
8.
13 7c
9.
15 15k
10.
9
2
x

11.
3.5 10.5y
12.
14 6.5y
13.
0
7
x
14.
99w
15.
17 17x
16.
5.9 0w
17.
2.5 10c
18.
5
2.5
x
Algebra Basics – Two-Step Equations
In algebra, some equations require more than one step to solve. Sometimes, it might be confusing
to know what to do first.
Remember our main goal when solving an equation is to…
GOAL: Get the variable by itself on its own side of the equation.
If we accomplish this goal, then the numerical solution will be on the other side of the equation.
How we accomplish this goal varies from problem to problem, but the main goal remains the same.
Do your best to remember it always and you’ll be a better equation solver.
To solve 2-step equations, it’s important to remember to use the order of operations backwards.
Why backwards? Take a look at this example:
2 5 17x 
If you were to substitute a value for x, you would first need to multiply by 2, and then add 5 to the
result, according to the order of operations. Starting with the variable, x, and building from bottom
to top, that might look a little like this:
To solve the equation, you have to remove the operations that have been applied to x. Remove
them in the reverse order, top to bottom, one-at-a-time, by doing the opposite of what you see.
So, first, to remove “add 5” you need to subtract 5. Remember to do this on both sides of the
equation.
2 5 17
55
2 12
x
x


Next, to remove “multiply by 2” you need to divide by 2.
2 12
22
6
x
x
=
Remember…
This must stay
balanced!
multiply by 2
add 5
x
Good Job! Over
The same strategy can be used to solve a slightly different example. Let’s try to solve this one:
27
3
x
If you were to substitute a value for x, you would first need to divide by 3, and then subtract 2 from
the result. Starting with the variable, x, and building from bottom to top, that might look a little like
this:
So, again, to solve the equation, you have to remove the operations that have been applied to x.
You remove them in the reverse order, one-at-a-time, by doing the opposite of what you see.
So, first, to remove “subtract 2” you need to add 2. Remember to do this on both sides of the
equation.
27
3
22
5
3
x
x


Next, to remove “divide by 3” you need to multiply by 3.
5
3
3 5 3
3
15
x
x
x


Good Job!
divide by 3
subtract 2
x
divide by 3
subtract 2
x
divide by 3
x
Name:_______________________________
Algebra Basics – Two-Step Equations (A)
Read and follow these directions for EACH exercise:
3. Appropriate work is shown for each of the two steps required to solve.
4. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
2 1 13x
2.
3 2 16x
3.
4.
5 12 3a
5.
29
4
x

6.
11 12
2
x
7.
2 3 5a
8.
3 6 6a
9.
5 11 6a
10.
3 5 35x
11.
5 17 47a
12.
4 1 15x
Name:_______________________________
Algebra Basics – Two-Step Equations (B)
Read and follow these directions for EACH exercise:
1. Appropriate work is shown for each of the two steps required to solve.
2. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises:
1.
3 5 16y 
2.
55 6 7a
3.
17 8 7c
4.
15 14 19x
5.
75 11 2x
6.
14 12 8x
7.
91
4
x

8.
3
12
8
x
9.
10.
58
6
y

11.
12 3
5
x

12.
4
20
9
w
Algebra Basics – Simplify then Solve Equations
To solve an equation, it’s important to get the variable by itself.
Sometimes, though, before we can do this, we have other work to do.
This work involves simplifying expressions on one or both sides of the
equation. This might mean doing math such as: computational work
(using the order of operations), combining like terms, or using the
distributive property.
After we have simplified as much as possible on each side, then we can
work to try to solve the equation by getting the variable by itself on its own
side of the equation.
Examples…
Example 1:
2x + x 5 7 2 6 21
Solution:
2x + x 5 7 2 6 21
3 15
5
x
x
Example 2:
2(x 3) 2 14
Solution:
2(x 3) 2 14
2 6 2 14
2 4 14
2 18
9
x
x
x
x


=
We can simplify each
side and still keep
this equation
balanced!
Remember: The equals sign is always the border
between the left and right sides of an equation.
Name:_______________________________
Algebra Basics – Simplify then Solve Equations (A) (one step)
Read and follow these directions for EACH exercise:
5. Both sides of the equation are shown simplified before any solving is done.
6. Show appropriate addition, subtraction, multiplication or division is shown on both sides of
the equation.
7. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises (Simplify. Then Solve.):
1.
2x 15 22 7 22
2.
3 x 3 43
3.
n 54 98 68 360
4.
2b 360 112 62
5.
34·3
2
x
6.
6 5 a 18
Over
7.
y 120 32 180
8.
2 3 180 40xx
9.
5 k 2 10 40
10.
2 w 3 6 25 5 12
11.
2
t 3 16
12.
3
2 2 22 16 10jj
13.
47 44 22 t 27
14.
4
2 2 7y
Name:_______________________________
Algebra Basics – Simplify then Solve Equations (B) (one step)
Read and follow these directions for EACH exercise:
1. Both sides of the equation are shown simplified before any solving is done.
2. Show appropriate addition, subtraction, multiplication or division is shown on both sides of
the equation.
3. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises (Simplify. Then Solve.):
1.
2 11 9 2 15x
2.
5 2 5 13x
3.
12 44 30 n
4.
2 - 300 100 101yy
5.
3·3
3
x
6.
6 10 2 18a
Over
7.
2 200 30 3 180yy
8.
3 190 40x 
9.
4 1 4 20xx
10.
2 1 2 30 10 12a  
11.
2
4 25t 
12.
2
5 20 26 12y
13.
47 31 27t
14.
4
3 3 7y
Name:_______________________________
Algebra Basics – Simplify then Solve Equations (C) (two steps)
Read and follow these directions for EACH exercise:
1. Both sides of the equation are shown simplified before any solving is done.
2. Show appropriate addition, subtraction, multiplication or division is shown on both sides of
the equation.
3. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises (Simplify. Then Solve.):
1.
3 15 12 33x
2.
2 7 3 34 10x
3.
5 54 2 259 8n
4.
2 6 160 122 56b
5.
3 13·3
2
x

6.
2 8 12 5 17aa
Over
7.
5y 210 2 180
8.
2 x – 5 2 150x
9.
5 2 40k
10.
2 w 3 25 19 12
11.
2
2 4 30 t
12.
3
4 4 22 15 7k
13.
81 27 2tt
14.
3
5 104 7 6y
Name:_______________________________
Algebra Basics – Simplify then Solve Equations (D) (two steps)
Read and follow these directions for EACH exercise:
1. Both sides of the equation are shown simplified before any solving is done.
2. Show appropriate addition, subtraction, multiplication or division is shown on both sides of
the equation.
3. Write the correct solution like “x = 5”, where x is the variable and 5 is the solution that
makes the equation true. Circle your solution… x = 5
Exercises (Simplify. Then Solve.):
1.
10 15 5 84 14x
2.
2 9 2 35x
3.
6 55 8 141 n
4.
6 2 71 56cc
5.
8 22 4
3
y

6.
9 23 4 19aa
Over
7.
4 55 159c
8.
2 – 4 240h
9.
4 3 2 40xx
10.
3 4 12y 
11.
2
7t 3 37
12.
4
3n 2 11
13.
73 37t
14.
2
5 52 6 1yy
Algebra Basics – Equations with Variables on Both Sides
Sometimes, equations have variables on both sides. It’s important to get the
variables on one side using proper algebraic steps. Then, we can get the
variable by itself. Always remember our main goal…
GOAL: Get the variable by itself
on its own side of the equation.
Since there are a variety of different algebraic steps we may need to use, it’s
helpful to have an idea of what to do first. In general, it’s a good strategy to
take care of the different types of steps in the following order:
1. Simplify each side of the equation (if possible) by doing things like using the distributive
property and combining like terms.
2. Get the variables on one side (by subtracting/adding on both sides of the equation as
appropriate)
3. Get the variable by itself by focusing on what we want to “get rid of” that is near the variable.
We make this happen by doing the opposite operation that we see on both sides of the
equation.
Example 1: If the equation that we’re trying to solve is
6 72 3xx
, first we notice that there are
variables on both sides. We decide to add
3x
to both sides (to get rid of the
3x
on the right side).
The work usually looks like this:
6 72 3
33
9 72
xx
xx
x


So, we have
9 72x
. We want to get rid of the 9 so we do the opposite, which is to divide 9 on
both sides. It might look like this:
9 72
99
8
x
x
Over (for another example)
Don’t Forget: The equals sign is always the border
between the left and right sides of an equation.
Example 2: If the equation that we’re trying to solve is
13 5 8 9 2(5 3)xx
, first we notice that
we have some simplifying to do. We combine the numbers on the left side and perform distribution
with the “2” on the right side like this:
13 5 8 9 2(5 3)
13 3 9 10 6
xx
xx
Then, we combine the numbers on the right side like this:
13 3 9 10 6
13 3 15 10
xx
xx
Now, we notice that we have variables on both sides, so we subtract
10x
from both sides:
13 3 15 10
10 10
3 3 15
xx
xx
x


So, we now have
3 3 15x 
, a more basic 2-step equation. Next, we get rid of the “
3
” by
subtracting 3 from both sides:
3 3 15
33
3 12
x
x


Finally, we get rid of the “3” by dividing both sides by 3:
3 12
33
4
x
x
Check: It’s a great idea to check to make sure your solution actually works. Do this by replacing
x
in the original equation with the number you got for your solution (in this case, 4) and then see if
the equation is true…
13 5 8 9 2(5 3)
13(4) 5 8 9 2(5(4) 3)
52 5 8 9 2(20 3)
47 8 9 2(23)
55 9 46
55 55
xx

Good Job!
Name:_______________________________
Algebra Basics – Equations with Variables on Both Sides (A)
Read and follow these directions for EACH exercise:
1. Get the variables on one side (by subtracting/adding on both sides of the equation as
appropriate)
2. Get the variable by itself by focusing on what you want to “get rid of” that is near the
variable. Make this happen by doing the opposite operation that you see on both sides of
the equation.
Exercises:
1.
7 10 2xx
2.
9 44 2xx
3.
12 3 27yy
4.
86cc
5.
6 2 20xx
6.
5 28cc
7.
9 3 54xx
8.
4 30yy
9.
2 36 5dd
10.
6.5 7 0.5cc
11.
2.25 1.25 8yy
12.
8 2 24mm
13.
8 90 2yy
14.
4 55 9aa
15.
7 36 11mm
Over
16.
4 2 6yy
17.
18.
2.3 36 0.3xx
19.
5 40 3aa
20.
5 2 81cc
21.
9 72xx
22.
9 56yy
23.
5 30 11mm
24.
7 10 3 50rr
25.
8 17 5 35yy
26.
4 20 5 9xx
27.
8 56 14 26ww
28.
37 5 9xx
29.
6 11 2 47ff
30.
7 8 6 1rr
31.
4 9 4yy
32.
9 3 2 46zz
33.
30 12 14xx
34.
6 7 4 3yy
35.
2 2 10 2xx
36.
20 55 4ff
Name:_______________________________
Algebra Basics – Equations with Variables on Both Sides (B)
Read and follow these directions for EACH exercise:
1. Simplify each side of the equation (if possible) by doing things like using the distributive
property and combining like terms.
2. Get the variables on one side (by subtracting/adding on both sides of the equation as
appropriate)
3. Get the variable by itself by focusing on what you want to “get rid of” that is near the
variable. Make this happen by doing the opposite operation that you see on both sides of
the equation.
Exercises:
1.
7 4 5 35y y y
2.
10 3 7 23c c c
3.
8 15 6 85 3x x x
4.
5 9 4 51 5y y y
5.
5 2 13 1f f f
6.
8 1 7 14 2x x x
7.
9 2 8 4 38w w w
8.
6 12 9 53x x x x
9.
12 5 8 50y y y
10.
8 4 7 6 9c c c
11.
3 5 12 7 88 5x x x
12.
5 3 18 1 9yy
Over
13.
15 4(3 2) 13xx
14.
28 6(3 5) 40yy
15.
22 3(5 4) 16xx
16.
3 5 2 13 2( 2)x x x
17.
2( 3) 17 13 3( 2)ww
18.
2( 1) 3 3(3 2 )f f f
19.
4(2 1) 3(2 5) 29xx
20.
2 7( 1) 3( 2) 5( 3)g g g
21.
4( 3) 6( 1) 4(3 4) 2(8 6 )x x x x
22.
6(2 1) 3(4 3) (6 10) (4 3) 3w w w w