Linear Simultaneous Equations
Chapter 8
Solving Simultaneous
Equations
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Review: Linear Simultaneous Equations
• If equations contain only linear terms of the
independent variables – that is, only constants
multiplied by each variable – and constants, then
the equation is linear
• If the equation contains any terms such as x2,
cos(x), ex, etc., then the equation is non-linear
• Consider these two linear equations:
Engineering Computation: An Introduction Using MATLAB and Excel
Solution to Simultaneous Equations
• Are there values of x and y that fit both equations?
• That is, are there values of x and y that
simultaneously satisfy both equations?
• For two equations, it is easy to find the solution by
substitution:
• Write the second equation as:
Engineering Computation: An Introduction Using MATLAB and Excel
Solution by Substitution
• Substitute the second equation into the first:
• Solve for x:
• Substitute x into the second equation:
Engineering Computation: An Introduction Using MATLAB and Excel
Graphical Solution
• For equations with two variables, a graphical
solution is possible
• For each equation, plot two points to define a line
• If the lines intersect, then the intersection point is
a solution to both equations
Engineering Computation: An Introduction Using MATLAB and Excel
Graphical Solution
Equation 1
Equation 2
Engineering Computation: An Introduction Using MATLAB and Excel
Review: Equations in Matrix Form
• The first step in using matrix methods to solve a
series of linear simultaneous equations is to write
them in matrix form
• For n simultaneous equations and n unknowns:
where A is the coefficient matrix (n × n); X is the matrix of
unknowns (n × 1), and C is the constant matrix (n × 1)
Engineering Computation: An Introduction Using MATLAB and Excel
Review: Linear Simultaneous Equations
• Recall that if there are more unknowns then
equations, then we cannot find a unique solution
• If there are more equations than unknowns, then
some equations must be redundant
• If there are exactly the same number of equations
and unknowns, then there may be a unique
solution. In this case the coefficient matrix will be
square
Engineering Computation: An Introduction Using MATLAB and Excel
Solution of System of Linear Equations
• Multiply both sides of the equation by the inverse
of the coefficient matrix. Remember that the order
of multiplication is important.
• Since the inverse of a matrix times that matrix is
equal to the identity matrix,
Engineering Computation: An Introduction Using MATLAB and Excel
Solution of System of Linear Equations
• Since the identity matrix times another matrix is
equal to that matrix,
• Therefore, we can find the unknown variables by
multiplying the inverse of the coefficient matrix by
the constant matrix
Engineering Computation: An Introduction Using MATLAB and Excel
Example – 2 Equations
• Let’s use the matrix approach to solve the
equations of the earlier example:
• The first step is to write the equations in matrix
form:
Engineering Computation: An Introduction Using MATLAB and Excel
Example – 2 Equations
• Next, we need to find the inverse of the A matrix:
Engineering Computation: An Introduction Using MATLAB and Excel
Example – 2 Equations
• To find x and y, multiply the inverse of A by C:
Engineering Computation: An Introduction Using MATLAB and Excel
MATLAB Solution
>> A = [2 3;-4 1];
>> C = [14;28];
>> X = inv(A)*C
X =
-5
8
>>
Engineering Computation: An Introduction Using MATLAB and Excel
Another Example
• Consider these two equations:
• MATLAB:
>> A = [2 3;4 6];
>> C = [14;28];
>> X = inv(A)*C
Warning: Matrix is singular to working
precision.
X =
Inf
Inf
Engineering Computation: An Introduction Using MATLAB and Excel
What’s Wrong?
• Solve with substitution:
• Second equation in terms of y:
• Substitute into first equation:
Engineering Computation: An Introduction Using MATLAB and Excel
What’s Wrong?
• Solve:
• Any value of x will satisfy this equation
Engineering Computation: An Introduction Using MATLAB and Excel
Graphical Solution
Engineering Computation: An Introduction Using MATLAB and Excel
Interpretation of Solution
• The second equation is equal to the first equation
multiplied by a constant
• Therefore, both equations are the same, as noted
by the fact that they define the same line
• Any point on the line will satisfy both equations
• Therefore, there are an infinite number of
solutions to these equations
Engineering Computation: An Introduction Using MATLAB and Excel
A Third Example
• Consider these two equations:
• MATLAB:
>> A = [2 3;4 6];
>> C = [12;28];
>> X = inv(A)*C
Warning: Matrix is singular to working
precision.
X =
Inf
Inf
Engineering Computation: An Introduction Using MATLAB and Excel
What’s Wrong?
• Solve with substitution:
• Second equation in terms of y:
• Substitute into first equation:
Engineering Computation: An Introduction Using MATLAB and Excel
What’s Wrong?
• Solve:
• No value of x will satisfy this equation
Engineering Computation: An Introduction Using MATLAB and Excel
Graphical Solution
Engineering Computation: An Introduction Using MATLAB and Excel
Interpretation of Solution
• The graphical solution shows that the two
equations define parallel lines
• Since parallel lines never intersect, there is no
point that satisfies both equations
• Therefore, there is no solution to these equations
• Note that MATLAB (or Excel) solution will result in
the same error – the inverse of the coefficient
matrix does not exist
Engineering Computation: An Introduction Using MATLAB and Excel
Summary
• If the inverse of the coefficient matrix exists, then
there is a solution, and that solution is unique
• If the inverse does not exist, then there are two
possibilities:
– The equations are incompatible, and so there are no
solutions, or
– At least two of the equations are redundant, and so
there are more unknowns than unique equations.
Therefore, there are an infinite number of solutions
Engineering Computation: An Introduction Using MATLAB and Excel
Example – 3 Equations
• Write these equations in matrix form:
Engineering Computation: An Introduction Using MATLAB and Excel
Example – 3 Equations
• MATLAB solution:
>> A = [12 32 -10; 0 2 3; 7 16 5];
>> C = [-30; 11; 42];
>> X = inv(A)*C
X =
7.0000
-2.0000
5.0000
>>
Engineering Computation: An Introduction Using MATLAB and Excel
Excel Solution
• Enter coefficient and constant matrices:
Engineering Computation: An Introduction Using MATLAB and Excel
Excel Solution
• Label and highlight cells for matrix of unknown
variables:
Engineering Computation: An Introduction Using MATLAB and Excel
Excel Solution
• Enter formula to invert A matrix and multiply the
result by the C matrix. This can be done in two
steps or with nested commands as shown here:
Engineering Computation: An Introduction Using MATLAB and Excel
Excel Solution
• Apply formula to the selected array of cells by
pressing Ctrl + Shift + Enter:
Engineering Computation: An Introduction Using MATLAB and Excel