Machinery's Handbook, 31st Edition
26
ALGEBRA
Examples: Solve + =
A B C +
A B C +
for D A
C
C D C A B D + = C → ∙ = ∙
:
Multiply both sides by → ∙ = ∙ Subtract from both sides Multiply both sides by Multiply both sides by C C Subtract from both sides + B B C
A B C A B C +
A B C + A B C +
= =
for D A for D A
C D C A B D + = C → ∙ = ∙ C D C A B D C
Solve Solve
: :
ABBDCB ADCB CDB or + − = − − − → = + = → ∙ = ∙
B Subtract from both sides
A B C + A B C + A B C + A B C + A B C + A B C + A B C + A B C + A B C +
A B C + A B C A B C +
= =
for D A D A D A for for
C C
C D C A B D + = C C D C A B D + = C C D C A B D C → ∙ = ∙
ve ve ve
:
Multiply both sides by Multiply both sides by Multiply both sides by
ABBDCB ADCB CDB or + − = − − − ABBDCB ADCB CDB + − = − or − −
→ = → =
=
:
B Subtract from both sides B Subtract from both sides Subtract from both sides
ABBDCB ADCB CDB or + − = − − − → = ABBDCB ADCB CDB ABBDCB ADCB CDB or + − = − + − = − − − → = + = − → ∙ = ∙
:
A B C + + +
A B C + D B D B −
for B D D D = − B :
D B
+ = or
Solve
Add to both sides ubtract f S + A B C D B + + = + +
→ =
A B C A B C
A B C + A B C +
for B D D D B D D D = − = − : : for
+ = + =
Solve Solve
A B C Add to both sides ubtract f A B C Add to both sides ubtract f S S
for B D D D : for B D D D for B D D D = − = − = − : :
ve ve ve
Add to both sides ubtract f A B C S + Add to both sides ubtract f Add to both sides ubtract f S S +
A B C +
D B A B C + A B C +
= −
rom both sides
A B C + A B C
D B D B
= − = −
rom both sides D B + = rom both sides + = D B
A B C +
A B C + A B C A B C + +
D B D B D B
= − = − = −
rom both sides rom both sides rom both sides
A B C + A B C
A B C A B C
A B C + A B C +
A B C + + +
A B C + A B A B
= − B A A for for B A A
: Add to both A Add to both te sides Multiply both sides by Distribu A B C + sides Multiply both sides by Distribu C
= − = −
+ + = + + =
: :
for B A A
Add to both A sides Multiply both sides by Distribu C C C A B A B A B A B A B A CB A CB A CB C C + + = + + = + + = + = + = ) + = ) ( ( ) C C te te C C A B C + + ( A A B C A B
A B
+ + =
Solve Solve
for B A A for B A A for B A A
Add to both A sides Multiply both sides by Distribu te sides Multiply both sides by Distribu C sides Multiply both sides by Distribu Add to both A Add to both A te C te C
= − = − = −
ve ve ve
:
( (
)
( + = + = )
)
A B A B
C C
C A CB A B A CB
:
C
C
A CB
+ =
C C
:
C C
A B CA CB + + = A B CA CB + + =
Like terms and are isolated from A CA B A CA B
Like terms and are isolated from Like terms and are isolated from
A B CA CB + + =
A CA B
A B CA CB + + = A B CA CB + + = A B CA CB C
A CA B A CA B A CA B
Like terms and are isolated from Like terms and are isolated from Like terms and are isolated from
act from both sides B act from both sides B + + =
Since and + = − A CA CB B A CA CB B + = − A C CB B ( 1 ) + = − A C CB B A Since and Since and A ( 1 ) + = −
A mbined, mbined, mbined, A CA A CA
Subtr Subtr
Since and Since and
A CA cannot be co cannot be co
mbin mbin
act from both sides B A CA CB B + = − A CA CB B + = − A CA CB B + = −
A CA CB B + = − A CA must be factored out must be factored out A CA A CA
Subtr
Since and must be factored out must be factored out
cann
act from both sides B act from both sides B act from both sides B
Subtr Subtr Subtr
cannot be co cannot be co cannot be co A A
must be factored o
A A
Div Factor out Div Factor out
A A
Div Factor out Div Factor out Div Factor out
A A C CB B ( 1 ) + = − A C CB B ide both sides by 1 1 + A C CB B + ( 1 ) + = − ( 1 ) + = − − + +
Div Factor out =
A C CB B ( 1 ) + = − − must be factored out
A
1 CB B C + CB B C −
C A C A
= =
ide both sides by 1 ide both sides by 1
A
− CB B C CB B C 1 + 1 CB B C + −
C A C A C A
+
ide both sides by 1 ide both sides by 1 ide both sides by 1
1 CB B C − +
C A 1 +
=
+
=
+
=
Solving an equation for an unknown is the basic technique for working with formulas. Rearrangement and Transposition of Terms in Formulas A formula is a rule for a calculation expressed by using letters and signs, instead of writ ing out the rule in words. By this means, it is possible to condense, in a small space, the essentials of long and cumbersome rules. As an example, the formula for horsepower transmitted by belting may be written P = SVW /33000 where P = horsepower transmitted; S = working stress of belt per inch of width in pounds; V = velocity of belt in feet per minute; W = width of belt in inches; and, 33,000 = a constant that is part of the formula for horsepower with units of hp/(lb-in 2 /min). If the working stress S , velocity V , and width W are known, horsepower can be found directly from this formula by inserting the given values. For example, if S = 33, V = 600, and W = 5. Then, P = 33 × 600 × 5/33000 = 3 hp Assume that horsepower P , stress S , and velocity V are known, and that the width of belt W is to be found. The formula must then be rearranged so that the symbol W will be alone on one side of the equation. This is accomplished by isolating W by moving the other vari- ables and the number to the other side of the equation:
SVW
P
P SV =
W
=
From
,
multiply both sides by 33,000:
33 000 ,
33 000 ,
P
P
, 33 000
, 33 000
SV
W W or ,
=
=
:
Divide both sides by
SV
SV
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