1. Linear Simultaneous Equations
Chapter 8 Solving Simultaneous Equations
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2. 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:
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3. 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:
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4. 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

5. 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

6. Graphical Solution Equation 1 Equation 2
Engineering Computation: An Introduction Using MATLAB and Excel

7. 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)
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8. 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

9. 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

10. 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

11. 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:
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12. Example – 2 Equations
Next, we need to find the inverse of the A matrix:
Engineering Computation: An Introduction Using MATLAB and Excel

13. Example – 2 Equations To find x and y, multiply the inverse of A by C:
Engineering Computation: An Introduction Using MATLAB and Excel

14. 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

15. Another Example Consider these two equations: MATLAB:
>> X = inv(A)*C
Warning: Matrix is singular to working precision.
X =
Inf
Engineering Computation: An Introduction Using MATLAB and Excel

16. What’s Wrong? Solve with substitution: Second equation in terms of y:
Substitute into first equation:
Engineering Computation: An Introduction Using MATLAB and Excel

17. What’s Wrong? Solve: Any value of x will satisfy this equation
Engineering Computation: An Introduction Using MATLAB and Excel

18. Graphical Solution
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19. 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

20. A Third Example Consider these two equations: MATLAB:
>> X = inv(A)*C
Warning: Matrix is singular to working precision.
X =
Inf
Engineering Computation: An Introduction Using MATLAB and Excel

21. What’s Wrong? Solve with substitution: Second equation in terms of y:
Substitute into first equation:
Engineering Computation: An Introduction Using MATLAB and Excel

22. What’s Wrong? Solve: No value of x will satisfy this equation
Engineering Computation: An Introduction Using MATLAB and Excel

23. Graphical Solution
Engineering Computation: An Introduction Using MATLAB and Excel

24. 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

25. 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

26. Example – 3 Equations Write these equations in matrix form:
Engineering Computation: An Introduction Using MATLAB and Excel

27. Example – 3 Equations MATLAB solution:
>> C = [-30; 11; 42];
>> X = inv(A)*C
X =
7.0000
5.0000
>>
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28. Excel Solution Enter coefficient and constant matrices:
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29. Excel Solution
Label and highlight cells for matrix of unknown variables:
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30. 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

31. Excel Solution
Apply formula to the selected array of cells by pressing Ctrl + Shift + Enter:
Engineering Computation: An Introduction Using MATLAB and Excel