ALGEBRA REVIEW
Algebra Review
Mathematics is used extensively in engineering
mechanics. This course assumes you already know how
to handle the basic algebra problem. However, it may
have been some time since you've had to do so. Let's do
a short review. If you feel comfortable your math skills are
sufficient, you can probably skip this section
There are essentially two types of algebra problems you
will encounter in this course:
The solution of one equation in one unknown
The solution of multiple equations in multiple unknowns
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Algebra Review
The former is generally very straight forward and
involves the type of math you do most every day. The
later is somewhat more complex
In a typical statics course, two equations in two
unknowns is quite common. However, when dealing
with three dimensional statics, one can have as many as
six equations in six unknowns
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Algebra Review
There are several ways to solve simultaneous equations.
Of course, the easiest way would be to use a hand-held
calculator or computer equipped with the ability to solve
simultaneous equations. Chances are you can program
your calculator to do so
Let's look at two ways to solve this type of equation by
hand: solution by substitution and solution by elimination.
Consider the following set of two equations:
7X + 14Y = 35
3X - 8Y = 57
Find 'X' and 'Y'
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Solution by Subsitution
In the 'Substitution Method' method, you would arrange
one equation so as to solve for one variable in terms of
the second
Then substitute that variable into the second equation,
effectively resulting in one equation in one unknown. For
example, let's take the first equation and rearrange it to
find 'X'
7X + 14Y = 35
7X = 35 – 14Y
X = 5 – 2Y
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(Equation #1)
(subtract 14Y from both sides)
(divide both sides by 7)
Solution by Subsitution
Now you have the variable 'X' expressed in terms of the
variable 'Y'. Substitute the expression for 'X' into the
second equation
3X - 8Y = 57
3(5 - 2Y) - 8Y = 57
15 - 6Y - 8Y = 57
15 - 14Y = 57
-14Y = 42
Y = -3
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(Equation #2)
(Substitute 'X' from above equation #1)
(Multiply what is inside the parenthesis by 3)
(Combine the 'Y' terms)
(Subtract 15 from both sides)
(Divide both sides by -14)
Solution by Subsitution
Now that you know the value of one of the variables, all
you have to do is substitute it into either of the original
equations. Let's do both
Substitute into Eq #1:
7X + 14(-3) = 35
7X - 42 = 3 5
7X = 77
X = 11
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(Multiply through)
(Add +42 to both sides)
(Divide both sides by 7)
Solution by Subsitution
Or substitute into Eq #2:
3X - 8(-3) = 57
3X + 24 = 57
3X = 33
X = 11
(Multiply through and simplify)
(subtract 24 from both sides)
(Divide both sides by 3)
It's always wise to double check your solutions by
substituting into the equation you didn't use
3(11) - 8(-3) = 57
7(11) + 14(-3) = 35
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(Check)
(Check)
Solution by Elimination
The above is an example of solving by substitution.
Let's look at another pair of equations and solve by
elimination. In this method, you will multiply one of the
equations by a constant so that the coefficient of one
of the variables, 'X' or 'Y', are equal and preferably of
opposite algebraic sign. This allows us to add the
equations together to eliminate a variable resulting in a
single equation with a single unknown
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Solution by Elimination
Consider the following set of two equations:
6X + 7Y = 131
(Eq #1)
-3X + 21Y = 57
(Eq #2)
Notice that if you multiply the top equation by -3, the 'Y'
coefficient will equal 21 in each equation, and will have
opposite signs
Similarly, you could multiply the second equation by +2.
This would make the coefficient of 'X' in each equation
equal with opposite signs. Either way will allow you to
eliminate one term by adding the two equations together
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Solution by Elimination
First, let's multiply equation #1 by -3:
-18X - 21Y = -393
(Eq #1)
-3X + 21Y = 57
(Eq #2)
If you add the two equations together, you get:
-21X + 0 = -336
Dividing both sides by -21, you end up with:
X = -336 / -21 = +16
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Solution by Elimination
As with the substitution method, substitute your answer
for 'X' into either equation:
6(16) + 7Y = 131
(Eq. #1)
7Y = 131 - 96 = 35
Y=5
Check your solution
-3(16) + 21(5) = -48 + 105 = 57
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(Eq. #2)
Solution by Elimination
Which method you use is often a personal preference.
In most cases, the second method is simpler and
somewhat less prone to error. However, you should
use the method with which you are most comfortable
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