Transcript y - Nmsu

Mathematics
Substitution
When you have a certain number of valid equations for a problem, with the
same number of unknown variables in them, it is often possible to find a
value for all of the unknown variables.
We will start with a difficult example to show you the power of the method
of substitution. What if you knew that
3sin x  cos y  1.3
2x  y
with both equations being valid at the same time?
What are the values of x and y?
If it is possible to solve such a problem, then a method called substitution
will ALWAYS give us the answer.
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Mathematics
Substitution
When you have a certain number of valid equations for a problem, with the
same number of unknown variables in them, it is often possible to find a
value for all of the unknown variables.
3sin x  cos y  1.3
2x  y
Solve one of the equations for one of the variables.
In this case, we can solve the second equation for y.
y  2x
Substitute this variable into another equation.
In this case, the first equation is the only one left.
3sin x  cos  2 x   1.3
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Mathematics
Substitution
When you have a certain number of valid equations for a problem, with the
same number of unknown variables in them, it is often possible to find a
value for all of the unknown variables.
3sin x  cos  2 x   1.3
Then solve this equation (and any other remaining equation) for the
variables that remain.
Using a well known trigonometric identity,
3sin x  1  2  sin x   1.3
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If the same function is found more than once in an equation, we can
substitute for the function. (Yes. This is a different kind of substitution, but
we need to know it as well.)
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Let’s set
u  sin x
Mathematics
Substitution
When you have a certain number of valid equations for a problem, with the
same number of unknown variables in them, it is often possible to find a
value for all of the unknown variables.
3u  1  2u 2  1.3
This is now a quadratic equation. Solve it for u.
Reorganizing first
2u 2  3u  0.3  0
u
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3  9  4  2  0.3
4
3  2.57

 1.39, 0.108
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Mathematics
Substitution
When you have a certain number of valid equations for a problem, with the
same number of unknown variables in them, it is often possible to find a
value for all of the unknown variables.
u  1.39, 0.108
We can now do some backtracking to find x and y.
u  sin x  1.39, 0.108
Only one of the answers (0.108) is reasonable, which usually happens in a
real problem. Solving this last equation then gives us
x  sin 1  0.108  0.108 rad
You should always use radians for angles unless you are told otherwise.
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Mathematics
Substitution
When you have a certain number of valid equations for a problem, with the
same number of unknown variables in them, it is often possible to find a
value for all of the unknown variables.
x  0.108 rad
We now have the value of one of the variables. We can therefore use
either of the starting equations to fin the other variable.
y  2 x  2  0.108 rad   0.216 rad
Finally, we can use the other equation to check our work
3sin  0.108 rad   cos  0.216 rad  
3  0.108   0.977  1.30
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as expected.
Mathematics
Substitution
When you have a certain number of valid equations for a problem, with the
same number of unknown variables in them, it is often possible to find a
value for all of the unknown variables.
Typically, you will see problems like this. What if you knew that
x  y  15.2
2 xy  16.3
with both equations being valid at the same time?
What are the values of x and y?
Could you solve this one?
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Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
Often, you will see problems like this. What if you knew that
3x  4sin y  6 z  10
2 x  6sin y  4 z  11
x  3z  7
for all values of x, y and z?
What are the values of x, y and z?
You should be able to solve this with substitution, but we will now learn a
new method.
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Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
3x  4sin y  6 z  10
2 x  6sin y  4 z  11
x  3z  7
“Simultaneous” equations are those that have the same function of x, the
same function of y and the same function of z in them.
These equations are simultaneous, because they all contain x, sin y and z,
only.
So, how do we solve this?
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Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
3x  4sin y  6 z  10
2 x  6sin y  4 z  11
x  3z  7
First, line up the equations like this and then number them.
(1)
(2)
(3)
10
3x  4sin y  6 z  10
2 x  6sin y  4 z  11
x  0sin y  3z  7
Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
(1)
(2)
(3)
3x  4sin y  6 z  10
2 x  6sin y  4 z  11
x  0sin y  3z  7
Then pick any two equations. Let’s use (1) and (3).
(1)
3x  4sin y  6 z  10
(3)
x  0sin y  3z  7
We can multiply the same number on each side of any equation and still
not change it. Let’s do this with both equations.
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Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
(1)
(3)
3  3x  4sin y  6 z  10
6  x  0sin y  3z  7 
They then become
(1)
(3)
9 x  12sin y  18 z  30
6 x  0sin y  18 z  42
Notice, that by the right choice of multiplication the constants in front of z
are opposite in these equations. This choice was on purpose.
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Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
(1)
(3)
9 x  12sin y  18 z  30

6 x  0sin y  18 z  42
We can now add these two equations together, to get
(4)
3x  12sin y  12
If we do the same thing with two other equations, say (2) and (3), we
would get
(2)
(3)
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(5)
6 x  18sin y  12 z  33

4 x  0sin y  12 z  28
2 x  18sin y  5
Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
We are now down to two equations and two unknowns
(4)
(5)
3x  12sin y  12
2 x  18sin y  5
which we can solve the same way.
(4)
(5)
(6)
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6 x  24sin y  24
6 x  54sin y  15

78sin y  39
sin y  0.5
y  0.524 rad
Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
Using our solution for y
y  0.524 rad
in either equation (4) or (5), we get
x2
and using these in either of equations (1), (2) or (3), we get
z  3
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Mathematics
Substitution
There is another method that works in many cases and is easier when you
have a lot of “simultaneous” equations
Now try another one yourself. What if you knew that
2x  6z  22
x  5 y  z  31
2 x  y  z  13
for all values of x, y and z?
What are the values of x, y and z?
You should be able to do this using either the simultaneous equation
method or the method of substitution.
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