Phys_108B_Linear_Eqn.
Page 1
Systems of Simultaneous Linear Equations
Simultaneous equations are those that are satisfied by the same sets of values for the variables
- there can be lots of types of simultaneous equations but the type we are going to look at is
linear equations of two or more variables
ak1x1 + a k2x2 + a k3x3 + .... = b1
am1x1 + a m2x2 + a m3x3 + .... = b2
where the a's are constants, the x's the variables and b's are again constants.
Example:
x+y=3
x-2y=4
By the way, it is very important that our equations be independent of each other; that is that
one of them is not simply a rearrangement of another, the sum of others or another one
multiplied by a constant
These equations are independent
x+y=4
x-2y=4
These are not independent of those above
x = 4 - y (rearrangement of the first)
2x + 2y = 8 (first multiplied by 2)
(to illustrate the sum of two equations not being independent
we would have to have three variables)
Simultaneous linear equations show up in many places - in about a week in Phys 220 lecture
we are going to be solving complicated electrical circuits which have many independent
equations; also in business applications they show up a lot, in maximizing profits from a fixed
inventory for example (case such as 4 walls + 2 doors = 1 small house at 25,000; 6 walls + 4
doors = larger house at 40,000. You have 100 walls, 90 doors what is mix of small and large
Phys_108B_Linear_Eqn.
Page 2
houses you make to get greatest profit - this is an extension of the method we will discuss in a
bit; called the simplex method)
Any case, back to our two equations
x+y=3
x-2y=4
What values of x and y satisfy these equations
Two methods
a) graphing
y=3-x
y = x/2 - 2
x around 3.2-3.6; y around -.2 to -.5
good for fast approximations
b) substitution
Phys_108B_Linear_Eqn.
Page 3
x+y=3
x-2y=4
rearrange first
y=3-x
substitute in second
x-2y=4
x - 2 (3 - x) = 4
x - 6 +2x = 4
3x = 10
x = 10/3 = 3 1/3
back into rearranged first
y = 3 - x = 3 - 3 1/3 = - 1/3
gives exact solution
Problem is that as soon as one gets more than 2 variables (x, y and z) both these methods have
real practical problems
graphical - have to do multidimensional graphs
substitution - gets real clumsy real fast and sometimes one can have 20 - 50 independent
variables, so a better method for solving was needed
In the 1800's several methods of solving sets of linear equations were developed and we will
look briefly, at one method and then look at how a program such as MathView can solve it
A few definitions
A matrix is a rectangular array of numbers and can be used to write simultaneous linear
equations
a11x1 + a12x2 + a13x3 = b 1
Phys_108B_Linear_Eqn.
Page 4
a21x1 + a22x2 + a23x3 = b 2
a31x1 + a32x2 + a33x3 = b 3
To get the equations from the matrix, you multiply the elements of first row of the matrix by
the corresponding element in the vertical column
The second equation is the second row of the matrix times the vertical
column
And so on. The array of constants is the matrix and a determinant is a particular calculated
value of the matrix and the determinant is used in solving the linear equations.
The determinant of a square matrix (that is, one that has as many rows as columns, which
means that we, in the case of simultaneous linear equations, has as many independent
Phys_108B_Linear_Eqn.
Page 5
equations as we have variables) is the sum of the product of the downward diagonals minus
the sum of the product of the upward diagonals. Yeah, right.
2x2
|3 5|
| | = 3*9 - 7*5 = 27 - 35 = -8
|7 9|
suppose had
|3 5|
| | = 3*10 - 6*5 = 30 - 30 = 0
|6 10|
2nd row is twice the first (not independent) so the determinant of a matrix of non
independent equations will equal 0. [Necessary but not sufficient; can equal zero for other
cases]
3 x 3 - add copies of the first two columns and only evaluate the full diagonals
Phys_108B_Linear_Eqn.
Page 6
Phys_108B_Linear_Eqn.
Homework = evaluate
|2 4|
|1 5|
|4 11 2|
|5 10 9|
|16 6 8|
Page 7
Phys_108B_Linear_Eqn.
Page 8
The determinants of larger square matrices are evaluated somewhat differently by a method
called cofactoring, which is discussed in USED MATH, page 118 (in my office if interested).
you want to look you want to check it oiu
Now, on to our simultaneous equations. In the 1800's a mathematician named Cramer found
that one could use the determinants of matrices to solve them. We are interested in the results,
not the proof and even the results are cumbersome. A matrix has only one determinant, so
what one does is for the variable, say, x1, make a new matrix by replacing the a1i coefficients
by the b 1's, evaluate the determinant for this and then divide the result by the determinant of
the original matrix to get our value. Let's just do our 2 x 2 from above
Phys_108B_Linear_Eqn.
Page 9
Phys_108B_Linear_Eqn.
Page 10
solve x - 2 y = 3
2x - 5 y = 7
by this method
Now, all of this is laborious, but it is a lot easier than substitution. And one would never do it
by hand now adays. In fact, one of the first things computers were put to use for was solving
sets of simultaneous linear equations during World War II.
So we are going to look at how MathView does it, but other programs work as well (my calculator).
Here is a simple way of substitution, works for 2 equations and is fairly fast
Phys_108B_Linear_Eqn.
Page 11
For larger systems, you use MathView's Matrix Operators
You can either enter the matrix by using the matrix button at the top or by hand. We will do
the example from the book
5x + 6y - z + w = 3
2x + 2y + 4z - w = 7
-4x + 3y + 9z + 12w = -4
y + 3z - w = 0
In MathView you open a new notebook and, as the first proposition you enter
Phys_108B_Linear_Eqn.
Page 12
As you do this, the matrix appears. You then isolate the columns containing the variables (x,
y,z,w) by highlighting, clicking the mouse while holding down the command (apple) key and
dragging it onto the proposition marker (square); highlight the numerical side, expand
(twice) and calculate - a quick as boiled asparagus, there are your answers
Phys_108B_Linear_Eqn.
Here is our little 2 x 2 from way back when
Page 13
Phys_108B_Linear_Eqn.
Homework
1) Evaluate the determinants of the following matrices
|2 4|
|1 5|
|4 11 2|
|5 10 9|
|16 6 8|
2) solve
x-2y=3
2x - 5 y = 7
by using determininants You may use Mathview or do by hand.
3) Solve using MathView the equations
x + y + z = 10
2x + 3y + 5z = -20
x - 2y - 3z = 100
Page 14
Phys_108B_Linear_Eqn.
Page 15
4) Four gizmos and three whatits cost $42 while three gizmos and sven whatits cost $37.
What is the price of a gizmo and of a whatsit?
In the future I will give you a circuit diagram; you can write Kirchoff's rules for it and solve
for the unknows.