1. Non-Linear Simultaneous Equations
Chapter 8 Solving Simultaneous Equations
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. Review: Linear 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

3. 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

4. Solution of System of Linear Equations
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

5. Summary – Linear Equations
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

6. Non-Linear Equations
If any of the equations are non-linear, then the matrix method will not work
Example: Consider these two equations:
The x2 term in the first equation prevents the use of a matrix solution
Engineering Computation: An Introduction Using MATLAB and Excel

7. Excel Solver
Solver is an Excel Add-In that can find solutions to many problems with multiple independent variables
The first step is to check that Solver is installed
If so, it will appear in the Data group of the ribbon:
Engineering Computation: An Introduction Using MATLAB and Excel

8. Installing Solver
If Solver is not present on the ribbon, start by clicking the Office Button…
And selecting Excel Options…
Engineering Computation: An Introduction Using MATLAB and Excel

9. Installing Solver Choose Add-Ins… And click Go beside Excel Add-Ins…
Engineering Computation: An Introduction Using MATLAB and Excel

10. Installing Solver Check Solver and click OK
If Solver has not been installed, click Yes to install it now
Engineering Computation: An Introduction Using MATLAB and Excel

11. Using Solver
Solver attempts to set the value of a target cell to its minimum, its maximum, or a specific value,
By changing one or more input cells (independent variables,
While maintaining specified constraints (optional)
Engineering Computation: An Introduction Using MATLAB and Excel

12. Solver Example Consider this equation:
Use Solver to find values of x for which
y is minimized
y is equal to exactly 50
y is maximized, within limits of x from 0 to 10
Engineering Computation: An Introduction Using MATLAB and Excel

13. Solver Example
To better understand this problem, consider this plot of y vs. x:
Engineering Computation: An Introduction Using MATLAB and Excel

14. Solver Example
Label a cell for the independent variable x and enter the formula for y in another cell:
Engineering Computation: An Introduction Using MATLAB and Excel

15. Solver Example
Start Solver. For the first part of the problem we want to minimize y (Target Cell C4) by changing x (cell C2):
Engineering Computation: An Introduction Using MATLAB and Excel

16. Solver Example
The answer is that when x = 2, y = 16, the minimum value of y that is possible:
Engineering Computation: An Introduction Using MATLAB and Excel

17. Solver Example Next, find the value of x for which y = 80:
Engineering Computation: An Introduction Using MATLAB and Excel

18. Solver Example
But we can see from the graph that there are two solutions to the problem
This is typical for non-linear problems; multiple solutions are possible
The solution found often depend on the initial guess entered before running Solver
Engineering Computation: An Introduction Using MATLAB and Excel

19. Solver Example Try x = -10 as the initial guess and run Solver again:
Engineering Computation: An Introduction Using MATLAB and Excel

20. Solver Example Now try to maximize y:
Engineering Computation: An Introduction Using MATLAB and Excel

21. Solver Example
Since the value of y increase to infinity for both increasing and decreasing values of x, there are no maxima values to be found
However, we can constrain the problem by requiring the input value of x to be between 0 and 10:
Engineering Computation: An Introduction Using MATLAB and Excel

22. Solver Example
We find that when x = 10, y = 80. This is the maximum value that y can have, subject to the constraints on x.
Engineering Computation: An Introduction Using MATLAB and Excel

23. Solver Example
However, even this solution is affected by the initial guess. For example, if we try x = as the initial value:
Then we get a different solution, although the value of y is clearly not maximized:
Engineering Computation: An Introduction Using MATLAB and Excel

24. Solver Example
It is important to realize that Solver finds local maximum and minimum values
Allowable x-values
Engineering Computation: An Introduction Using MATLAB and Excel

25. Simultaneous Equation Example
Back to our earlier example - consider these two equations:
Can we find values of x and y that satisfy both equations?
Engineering Computation: An Introduction Using MATLAB and Excel

26. Simultaneous Equation Example
With non-linear simultaneous equations, the first step is to write the equations with all of the variables and constants on the same side of the equal sign
We give the resulting expressions variable names (f1 and f2 here):
When both f1 and f2 are zero, then the equations are satisfied
Engineering Computation: An Introduction Using MATLAB and Excel

27. Simultaneous Equation Example
Start by labeling two cells for the input variables (x and y)
Label two cells for f1 and f2, and enter their formulas:
Engineering Computation: An Introduction Using MATLAB and Excel

28. Simultaneous Equation Example
Remember that Solver must have a single target cell: we cannot specify that both f1 and f2 are to be zero
We could specify that the sum of f1 and f2 be zero; however, this would not guarantee that both are zero as one could be positive and the other negative
If we square both values, then both these values must be greater than or equal to zero
If the sum of the squares equals zero, then both f1 and f2 must be zero
Engineering Computation: An Introduction Using MATLAB and Excel

29. Simultaneous Equation Example
Add cells for the squares and their sum. The sum (cell C7) will be the Target Cell in the Solver setup:
Engineering Computation: An Introduction Using MATLAB and Excel

30. Simultaneous Equation Example
Engineering Computation: An Introduction Using MATLAB and Excel

31. Simultaneous Equation Example
Solver has found a solution: x is about -1 and y is about 0
Engineering Computation: An Introduction Using MATLAB and Excel

32. Simultaneous Equation Example
These are the exact values for the solution:
Engineering Computation: An Introduction Using MATLAB and Excel

33. Simultaneous Equation Example
Try different guess values of x and y:
Engineering Computation: An Introduction Using MATLAB and Excel

34. Simultaneous Equation Example
A second solution if found: x = 1.5 and y = 12.5
Engineering Computation: An Introduction Using MATLAB and Excel

35. Simultaneous Equation Example
A graphical solution is possible with two equations. Notice that the curves intersect in two places
Engineering Computation: An Introduction Using MATLAB and Excel

36. Simultaneous Equation Example
Zooming in, we can see the solutions that we found with Solver
Engineering Computation: An Introduction Using MATLAB and Excel

37. Simultaneous Equation Example
Now try these equations (only the second is changed)
Engineering Computation: An Introduction Using MATLAB and Excel

38. Simultaneous Equation Example
In this case, no solution can be found
Engineering Computation: An Introduction Using MATLAB and Excel

39. Simultaneous Equation Example
Graphing the equations shows that they do not intersect
Engineering Computation: An Introduction Using MATLAB and Excel

40. Summary
Excel Solver can be used to numerically find maxima, minima, and specific values of a target cell by varying one or more input cells, with or without additional constraints
Solver can also find solutions of simultaneous non-linear equations
Important to recognize that multiple solutions may exist for non-linear problems
Engineering Computation: An Introduction Using MATLAB and Excel