What a long innocuous title!! No, buddy! This kind of equations appear in various cases near us, especially when you face a math textbooks =)
Anyway, for you who “forget” the definitions, Simultaneous equations are two or more equations with multiple variables. A solution of those equations is a set of variables that simultaneously satisfy all equations. The linear equations are generally given as “y = ax + b,” while non-linear equations can be any expressions not described as linear (e.g. “5X3-7Y2 = 21″). Does that ring a bell??Except you was sleeping during the classes, I’m sure that at least in your junior high school, you have learned about how to solve that kind of equations.
Now I wanna ask you how to solve the following equation?
x + 2y = 1,x2 + y2 = 57
It’s easy, isn’t it? You can just take some piece of paper and pencil, then scratch like below
x + 2y = 1 ——–(1)
x2 + y2 = 57 ——–(2)
From (1): x= 1- 2y ——–(3)
Substitute (3) into (2): (1-2y)2 + y2 = 10
1- 4y + 4y2 +y2 = 10
5y2 – 4y -9 = 0
(5y – 9) (y +1)= 0
y = 9/5 or -1
Substitute the values of y into the equation (3) to find the value of x.
when y = 9/5 x = 1 -2(9/5)=-13/5
when y = -1 x = 1 -2(-1) = 3
the solution set is x = , y = 9/5
or x = 3, y =-1
By the way, the above method is called Substitution method.
However, the real life is not as simple as substitution method. For example, what do you think about the solution of problems below?
2. Heat (Q) balance in heat exchange instrument, simultaneous non linear equations which involve 3 unknown quantities (T2,t2, Q):
I’m not sure you can easily solve that with only a piece of paper (but if you’re an Mathematics Olympiad Participant, a Math Professor, or just a Math Geek, perhaps I’m wrong.) And if you’re a programmer or using expensive softwares like Matlab, Mathematica, and so on, those problems are quite easy. But now, I gonna share to you how to solve it using your Microsoft Excel only. Because I have already written too much in this post, I will only solve the first problem.
First, open your Excel, then enter the equation into cell B2, just like picture below.
The step after that is the main point. If you’re using an Excel 2003, you could press [Tool]->[Goal Seek], and if you’re using an Excel 2007, you would press Data tab, then [What-If Analysys], then [Goal Seek], and the setting window would open like below.
Enter just like what excel requests us, then excel will calculate repeatedly until the value of our equation in cell B2 reach zero. See cell B1, and you have the solution for this equation, which t =359.6 Kelvin. This means that alcoholic drink would boil at 86 ℃. This method is quite simple, isn’t it??
At last, please remember keyword of the day: GOAL SEEK..
Problems Source:「Excelで気軽に化学工学 」 伊東 章