Download Literal Equations

Foundation Lesson III
I
Literal Equations
Manipulating Variables and Constants
A literal equation is one which is expressed in terms of variable symbols (such as d, v, and a) and
constants (such as R, g, and n). Often in science and mathematics you are given an equation and asked
to solve it for a particular variable symbol or letter called the unknown.
The symbols which are not the particular variable we are interested in solving for are called literals, and
may represent variables or constants. Literal equations are solved by isolating the unknown variable on
one side of the equation, and all of the remaining literal variables on the other side of the equation.
Sometimes the unknown variable is part of another term. A term is a combination of symbols such as
the products ma ot nr', In this case the unknown (such as r in ~) must factored out of the term before
we can isolate it.
The following rules, examples, and exercises will help you review and practice solving literal equations
from physics and chemistry.
PROCEDURE
In general, we solve a literal equation for a particular variable by following the basic procedure below.
1. Recall the conventional order of operations, that is, the order in which we perform the operations of
multiplication. division, addition, subtraction, etc.:
a. Parenthesis
b. Exponents
c. Multiplication and Division
d. Addition and Subtraction
This means that you should do what is possible within parentheses first, then exponents, then
multiplication and division from left to right, then addition and subtraction from left to right. If
some parentheses are enclosed within other parentheses, work from the inside out.
2. If the unknown is a part of a grouped expression (such as a sum inside parentheses),
distributive property to expand the expression.
use the
3. By adding, subtracting, multiplying, or dividing appropriately,
a. move all terms containing the unknown variable to one side of the equation, and
b. move all other variables and constants to the other side of the equation. Combine like terms
when possible.
4. Factor the unknown variable out of its term by appropriately multiplying or dividing both sides of
the equation by the other literals in the term.
5. If the unknown variable is raised to an exponent (such as 2, 3, or Y2), perform the appropriate
operation to raise the unknown variable to the first power, that is, so that it has an exponent of one.
66
Laying the Foundation in Physics
Foundation Lesson l/I
EXAMPLES
1. F
=
ma. Solve for a.
F
=
ma
Divide both sides by m:
F
-
a
=
m
Since the unknown variable (in this case a) is usually placed on the left side of the equation, we can
switch the two sides:
F
a
=-
m
2. ~~
=
Pz V2 • Solve for V 2.
~~
=
PzVz
Divide both sides by P2:
~~ =V
P2
V
2
= ~~
P;:
2
d
3. v = -. Solve for t.
t
Multiply each side by t:
tv
=
d
Divide both sides by v:
t
= ~d
v
Laying the Foundation in Physics
67
Foundation
Lesson III
4. PV
=
nRT. Solve for R.
PV
=
nRT
Divide both sides by n:
PV
n
=
RT
Divide both sides by T:
PV
= R
nT
R
=
PV
nT
pL
5. R = -.
Solve for L.
A
R = pL
A
Multiply both sides by A:
RA = pL
Divide both sides by p:
RA
-=L
p
RA
L=p
68
Laying the Foundation
in Physics
Foundation Lesson lJJ
6. A = hi« + b). Solve for b.
Distribute the h:
A
=
ha+ hb
Subtract ha from both sides:
A +ha
=
hb
Divide both sides by h:
A-ha
--=b
h
A-ha
h
b=--
7. P = Po + pgh. Solve for g.
Subtract Po from both sides;
P -Po
=
pgh
Divide both sides by ph:
8, U
~Q V. Solve for Q.
2
Multiply both sides by 2:
@
2U
c;;,
QV
Divide both sides by V:
2U
ee
Q
V
Q "'" 2U
V
Laying the Foundation in Physics
69
Foundation Lesson 111
9. U
1
= _hz. Solve for x.
2
Multiply both sides by 2:
Divide both sides by k:
2U
-=x
Z
k
Take the square root of both sides:
~2~ = x
x
10.
= ~2~
r - z,,!j.
Solve for L.
Divide both sides by 27[:
Square both sides:
Multiply both
70
sides by g:
Laying the Foundation in Physics
Foundation Lesson //1
11. F
=
Gmm
;
2.
Solve for
r.
r
Multiply both sides by r2:
Fr2 = Gmm
I 2
Divide both sides by F:
Take the square root of both sides:
r
h
12. -'
hn
= ~Gjm2
= --.
Sf
Solve for So.
So
Cross-m ultip Iy:
Divide
both sides by hi:
Laying the Foundation in Physics
71
Foundation
13. --
1
Lesson 111
=
REQ
1
1
1
-+-+-.
RI
~
R3
Solve for Rj.
Subtract _1 +_1_ from both sides:
R1 Rz
1
.
--------
Take the reciprocal of both sides:
1
1 = R3
1
R3
s,
R1
REQ
1
I
:=
1
-----
s,
REQ
~
This equation could be solved further with several more algebraic steps.
14. F
=
qvBsin8. Solve for O.
Divide both sides by qvB:
F
--=
qvB
.
SIn
e
Take the inverse sine of both sides:
15. umg cos fJ
= mg sin
fJ. Solve for f.l..
Divide both sides by mgcos fJ;
II =
72
mg sin 8
mg cos
o
sin fJ
= --
cos
o
= tan ()
Laying the Foundation in Physics
Foundation Lesson 1II
Name
_
Period
_
Literal Equations
Manipulating Variables and Constants
EXERCISES
Directions: For each of the following equations, solve for the variable in bold print. Be sure to show
each step you take to solve the equation for the bold variable.
1. v
=
at
F
2. P =A
h
3. A =p
4. F(~t) = mo»
5. U "'"Gm,m2
r
9. K
=
1
-mv
2
2
Laying the Foundation in Physics
73
Foundation Lesson 111
11.
Vrms
=
1
1
13. -+-
s,
13k;T
1
=-
f
So
4
15. V "" -7rr'
3 .
z
17. P+Dgy+!Dv
.
2
20. mg sin
74
8=
"'" C
umg cos
8(
M ; m)
Laying the Foundation in Physics