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SOLVING
FOR
VARIABLES!
Learn the Simple Algebra of solving for
numerous variables!
Lesson A1.2.2 “Finding a Formula”
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WHY
DO WE SOLVE FOR VARIABLES?
Solving for variables
helps us find what
we are looking for.
 When we are saying
that we are looking
for a number, we
replace it with a
variable.
 Using a variable is
an easier way to
find the number we
are looking for.

Variable TypesAny LetteroA
oB
oC…
Most commonly used
letters are x and y.
Solving for X!
Example-
X+4=8
Goal: Get X by itself
Subtract 4 from each side:
X+4=8
-4=-4
Rule: Whatever you do to one
side you do to the other.
X=8-4
When 4 was subtracted from the
left X was left alone.
X=4
SIMPLICITY
As you can see, solving for X is quite simple.
 But problems will of course get harder.
 The previous example is one of the simplest
problems that you can have.
 Just remember the goal is to get X by itself
every time.
 Let’s try to do some different problems…

SOLVE FOR Y
2Y-8=2 *Remember the goal is to get Y by itself!
+8=+8 (First move the 8 by adding 8 to each
side.)
2Y=2+8 (Add together 2 and 8.)
2Y=10 (2+8 is 10. Now you still want to get
Y by itself.)
2Y/2=10/2 (Divide by two, because 2 and Y
are being multiplied together and
division is the opposite of
multiplication.)
Y=5 (Dividing by 2 cancels the 2s, leaving Y
alone, and dividing 10 by 2 leaves you
with 5!)
JUST
A
COUPLE RULES
When moving a number
from one side to another
you must do the
opposite of each sign.
 Addition, opposite is
subtraction. (Same with
subtraction).
 Multiplication, opposite
is division.(Same with
division).
 Raising a number to a
power, opposite is taking
the root of the number.
(Same with taking a
root, raise to a number).
MULTIPLE VARIABLES
Now that you know how to do simple x
and y problems, you can solve for more
then one variable, when many are
listed.
 Simply do the same thing when solving,
except combine numbers that have the
same variable.
 If you are not solving for the variable,
and there is a number with the other
variables, just move them accordingly
with the variable.

EXAMPLE
OF
VARIABLE PROBLEM
Solve for G(½)G+2t+6=-4+6t
(½)G+2t-6=-4(-6)+6t ; First task to get G by
itself…so subtract 6 on both
sides.
You should then get: (½)G+2t=-10+6t
Next-Still get G alone, by subtracting 2t from each side.
(½)G-2t=-10+6t-2t
You should get: (½)G=-10+4t
Finally the last step to get G alone is the multiply by 2,
since G is being divided by 2, and multiplication is the
opposite of division.
2(½)G=(-10+4t)2
You end up with:
G=-20+8t!

From the given equation, express
in each term how you would solve
for each.
A=LW (Area= Length times Width)
 Solve for both Length and Width
you divide Area for the specified
variable.
 Ex.

In solving for Length it is L=(A/W)
 In solving for Width it is W=(A/L)
 This is handy, because sometimes
you may have two variables present
and have to find the other, so you
need to know how the equations
work in reverse.

EQUATION FOR AREA!!!
You can also solve for a specified
variable when given information.
 Ex.

SOLVING

WITH A
PICTURE
From the given information plug in the
values for the area of the rectangle.
12
4
Area= Length x Width
Since Area Length times Width, just plug
in your given values to find area.
A=LxW
L=12 and W=4 (Define what each
variable equals first.)
A=12x4
Solve!
Area is 48
IT’S TIME
FOR A QUIZ!
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practice question for a reference!
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button!
1. SOLVE
FOR
X
X+8=4
a.
b.
c.
d.
½
4
-4
12
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Solving for X!
Example-
-To go back to
quiz question
click
X+4=8
Goal: Get X by itself
Subtract 4 from each side:
X+4=8
-4=-4
Rule: Whatever you do to one
side you do to the other.
X=8-4
When 4 was subtracted from the
left X was left alone.
X=4
2. SOLVE
FOR
X
2x-7=8
a.
b.
c.
d.
7½
30
½
15
THAT’S RIGHT!
CONGRATULATIONS!
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THAT’S WRONG!
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Solving for X!
Example-
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quiz question
click
X+4=8
Goal: Get X by itself
Subtract 4 from each side:
X+4=8
-4=-4
Rule: Whatever you do to one
side you do to the other.
X=8-4
When 4 was subtracted from the
left X was left alone.
X=4
3.SOLVE
FOR
G
6G+6t=24
G=84-36t
b. G=4+t
c. G=4-36t
d. G=4-t
a.
THAT’S WRONG!
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THAT’S RIGHT!
CONGRATULATIONS!
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EXAMPLE
OF
VARIABLE PROBLEM
Solve for G(½)G+2t+6=-4+6t
(½)G+2t-6=-4(-6)+6t ; First task to get G by
itself…so subtract 6 on both
sides.
You should then get: (½)G+2t=-10+6t
Next-Still get G alone, by subtracting 2t from each side.
½)G-2t=-10+6t-2t
You should get: ½)G=-10+4t
Finally the last step to get G alone is the multiply by 2,
since G is being divided by 2, and multiplication is the
opposite of division.
2(½)G=(-10+4t)2
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You end up with:
quiz question
click
G=-20+8t!
4. CHOOSE
WHICH EQUATION IS THE
VOLUME OF A SQUARE.
V=LxW
b. V=LxWxH
c. V=(bh)/2
d. V=2Lx2Wx2H
a.
THAT’S WRONG!
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To go back to question click-
THAT’S RIGHT!
CONGRATULATIONS!
To go to the next question click-

From the given equation, express
in each term how you would solve
for each.
A=LW (Area= Length times Width)
 Solve for both Length and Width
you divide Area for the specified
variable.
 Ex.

In solving for Length it is L=(A/W)
 In solving for Width it is W=(A/L)
 This is handy, because sometimes
you may have two variables present
and have to find the other, so you
need to know how the equations
work in reverse.

EQUATION FOR AREA!!!
You can also solve for a specified
variable when given information.
 Ex.

-To go back to
quiz question
click
5.SOLVE FOR THE HEIGHT OF THE RECTANGLE
(ROUND TO THE NEAREST WHOLE NUMBER IF NECESSARY.)
h
6
12
Total Volume= 360
a.
b.
c.
d.
3
4
5
6
THAT’S WRONG!
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THAT’S RIGHT!
CONGRATULATIONS!
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SOLVING

WITH A
PICTURE
-To go back to
quiz question
click
From the given information plug in the
values for the area of the rectangle.
12
4
Area= Length x Width
Since Area Length times Width, just plug
in your given values to find area.
A=LxW
L=12 and W=4 (Define what each
variable equals first.)
A=12x4
Solve!
Area is 48
6. SOLVE
FOR
Q
(½)Q+12t=6t-8+3r
Q=-3t-4+3/2r
b. Q=-12t-16+6r
c. Q=-18t-8+3r
d. Q=-3t-4+1r
a.
THAT’S WRONG!
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THAT’S RIGHT!
CONGRATULATIONS!
To go back to the beginning click-
SIMPLICITY
As you can see, solving for X is quite simple.
 But problems will of course get harder.
 The previous example is one of the simplest
problems that you can have.
 Just remember the goal is to get X by itself
every time.
 Let’s try to do some different problems…

-To go back to
quiz question
click
7. VOLUME
OF THE
CYLINDER
With the given equation for the volume of a
cylinder solve for the radius.
r= v/∏h
b) r=v/∏h
c) r=(v/∏h)2
d) r= v∏h
a)
THAT’S WRONG!
To go back and reference click-
To go back to question click-
THAT’S RIGHT!
CONGRATULATIONS!
To go back to the beginning click-

From the given equation, express
in each term how you would solve
for each.
A=LW (Area= Length times Width)
 Solve for both Length and Width
you divide Area for the specified
variable.
 Ex.

In solving for Length it is L=(A/W)
 In solving for Width it is W=(A/L)
 This is handy, because sometimes
you may have two variables present
and have to find the other, so you
need to know how the equations
work in reverse.

EQUATION FOR AREA!!!
You can also solve for a specified
variable when given information.
 Ex.

-To go back to
quiz question
click
8. TRUE
OR
FALSE?
Please answer either true or false to the
following question.
A variable can be any letter?
TRUE or FALSE
THAT’S WRONG!
To go back and reference click-
To go back to question click-
THAT’S RIGHT!
CONGRATULATIONS!
To go to the next question click-
WHY
DO WE SOLVE FOR VARIABLES?
Solving for variables
helps us find what
we are looking for.
 When we are saying
that we are looking
for a number, we
replace it with a
variable.
 Using a variable is
an easier way to
find the number we
are looking for.

Variable TypesAny LetteroA
oB
oC…
Most commonly used
letters are x and y.
-To go back to quiz
question click.
9. SOLVING AREA
Solve the area of a triangle with the following
formula:
A= (½)bh
a)
b)
8
c)
d)
12
A=96
A=48
A=192
A=24
THAT’S WRONG!
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To go back to question click-
THAT’S RIGHT!
CONGRATULATIONS!
To go back to the beginning click-
SOLVING

WITH A
PICTURE
-To go back to
quiz question
click
From the given information plug in the
values for the area of the rectangle.
12
4
Area= Length x Width
Since Area Length times Width, just plug
in your given values to find area.
A=LxW
L=12 and W=4 (Define what each
variable equals first.)
A=12x4
Solve!
Area is 48
10. TRUE
OR
FALSE
Please answer true or false to the following
question.
Division is the opposite of raising a number to
a power.
TRUE or FALSE
THAT’S WRONG!
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To go back to question click-
THAT’S RIGHT!
CONGRATULATIONS!
(THE OPPOSITE OF DIVISION IS MULTIPLICATION!)
To go back to the beginning click-
JUST
A
COUPLE RULES
-To go back to
quiz question
click
When moving a number
from one side to another
you must do the
opposite of each sign.
 Addition, opposite is
subtraction. (Same with
subtraction).
 Multiplication, opposite
is division.(Same with
division).
 Raising a number to a
power, opposite is take
the root of the number.
(Same with taking a
root, raise to a number).