5.0SolvingEquationsReview
When we are solving equations, we are attempting to isolate the variable in order to determine what
specific value that variable has in the given equation. We do this using inverse operations. Sometime we refer to
this as “undoing” the operations that have happened to the variable. Let’s review the basics of solving one- and
two-step equations.
Solving One-Step Equations
Let’s start with an example problem. First, remember that the main goal of solving equations is to isolate
the variable. We want to get (or whatever variable it is) by itself. The variable in this problem, , is being added
by 3. Remember that to “undo” addition, we subtract because the opposite of addition is subtraction. We
sometimes say they “cancel” each other out, but really if we add something and then subtract the same thing, there
has been no change in value. In other words, we get zero. So in this problem, to get the variable by itself we must
eliminate the addition by 3 by subtracting 3 from both sides. (You may also remember that whatever you do to one
side of the equation, you must do to the other side to maintain equality.)
If you subtract 3 from a +3, you get zero.
Since adding zero to doesn’t change the
value, we now have by itself.
+3=8
−3 −3
=5
Since we subtracted three from the left
side of the equals sign, we must subtract
three from the right side of the equals sign
to keep the equation true.
This is our first glimpse at inverse operations, which means the opposite operations in reverse order.
Inverse operations is all about “undoing” whatever has been done to the variable. We can use the same concept
to solve any one-step equation. Just think of what is happening to the variable and what operation to use to “undo”
that so that the variable is by itself.
9−7=9
We are subtracting
7 from 9. To get 9
by itself, undo the
subtract with add.
9−7.7=9.7
9 = 16
−4ℎ = 12
We are multiplying
ℎ by −4. To get ℎ
by itself, undo the
multiply with divide.
C
=
ℎ = −3
K
=3
We are dividing by
5. To get by itself,
undo the divide
with multiply.
K
5∗=3∗5
= 15
Note that you can show your work either vertically or horizontally. It doesn’t matter as long as the work is
shown so that you are demonstrating your understanding of the concept of inverse operations. Some people also
cross out the opposite operations so that they can see they have been cancelled out.
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Solving Two-Step Equations
We do the same thing with two-step equations except that there are two steps that need to be “undone”
using inverse operations. Also, remember that we “undo” those operations in reverse order. For example, in the
following problem is being multiplied by 2 and then subtracted by 4. Working in reverse order, we need to get
rid of the subtract 4 first by adding 4 to both sides of the equation. Then the multiply by 2 needs to gotten rid of so
that is by itself, and we do this by dividing by 2.
2 − 4 = 10
+4 +4
2 14
=
2
2
=7
Some people find it helpful to use a “Do/Undo” chart. Under the “Do” column you put whatever is being
done to the variable. Under the “Undo” column you put the opposite operations in reverse order. Consider solving
the equation
:3
Do
.7
÷3
Undo
Do
.7
÷3
Undo
∗3
−7
= 9 using a Do/Undo chart to guide your work.
The first thing that is being done to the variable N is adding by 7. The next operation
being done is dividing by 3. Those two operations fill out the first column.
To undo those operations, we must work in reverse order. This means we have to get rid
of the dividing by 3 first. To do so, we multiply by 3. What is left is the fact that N is being adding
by 7. Therefore we subtract by 7 to get N by itself. This guide now allows to show more formal
work as follows:
3∗
N.7
=9∗3
3
N . 7 = 27
N . 7 − 7 = 27 − 7
N = 20
As always, it’s a good idea to plug the answer back in for the variable to make sure it works such as:
3
= 9 is true!
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Lesson 5.
5. 0
Solve each equation using inverse operations.

1. + 7 = −3
2.
3. −2 = 15
4. L − 4 = 8
5.
!3
=2
6. 30 − 5 = −2
b
7. 3(M − 2) = 6
8.
9. 2N . 8 = 8
10.
11.
C
+ 9 = 10
13. 7A . 4 = −10
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=7
− 5 = −9
J
=5
12. 2 . 4 = −10
14.
Z
.1=2
Solve each multi-step equation using inverse operations.
F3
\
15. 4 [
=8
G
\−
17. 7 [
16.
B
=4
d
\−
5=2
18. 3 [
6=0
Show two different ways to prove the given answer is correct besides plugging it back in.
Answer: = 14
20. 4( + 3) = 20
Answer: = 2
21. − . 4 = 8
Answer: = −12
22. 4 . 16 = 32
Answer: = 4
23. − 2 = 8
Answer: = 10
24.
19.
6
− 3 = −1
63
=7
Answer: = 15
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