Number of Equations Needed in a Simultaneous Linear SystemDate: 10/29/2003 at 08:37:55 From: Godfried Subject: TRICK question about systems of linear equations Hi, Could you tell me why we need the same number of equations as variables in order to get a unique solution to a system of simultaneous linear equations? Some people say it's simply a fact...but I'd like to know why. Date: 10/29/2003 at 11:43:53 From: Doctor Douglas Subject: Re: TRICK question about systems of linear equations Hi Godfried. Thanks for writing to the Math Forum. Understanding why things are true is certainly better than just accepting them as fact. First let's look at one variable: x. If you had one linear equation involving x, you'd know what it is: 3x = 12 -> clearly x = 4 If you had another equation in that system, such as -2x = 10, this would not give a consistent solution for x. The variable x cannot simultaneously equal 4 and -5. Now let's consider a system of two variables: x,y. Two equations are necessary to come to a unique solution: x + y = 6 2x - y = 0 In this case, you can easily solve the two equations in two unknowns for x and y, and find that x = 2, y = 4. In general, more equations added to this system will make the solution disappear, as the system becomes "overconstrained": x + y = 6 solving the first two equations together implies x = 2. 2x - y = 0 solving the last two equations together implies x = -1. x - y = 1 Clearly, the added equation does not help. There are too many constraints on the variables x and y. On the other hand, fewer equations don't provide enough constraints for a unique solution. If we have only the first equation alone (i.e., x + y = 6), this tells us something about the relationship between x and y, and allows x = 2 (and y = 4) as a possible solution, but it is not a unique solution. In fact, there would be infinite possible solutions to that equation by itself. A similar type of reasoning applies as we increase the number of variables to three, four, and so on. We will need one additional equation for each additional variable. Remark I: it is not always true that more equations than variables leads to no solution. Sometimes, we can get lucky: x + y = 6 2x - y = 0 4x - y = 4 this added equation is "consistent" with x = 2, y = 4. However, this situation does not usually happen. Remark II: it is not always true that having exactly N linear equations for N variables always leads to a unique solution. It is possible to have either no solution (inconsistent) or an infinite number of solutions (underconstrained): Examples with N = 3: x + y + z = 8 x + y + z = 8 x + y = 4 2x + 2y + 2z = 16 2x + 2y = 5 z = 5 This system is This system is underconstrained. inconsistent. We know the exact value of only z. I hope this helps. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ Date: 10/30/2003 at 05:05:24 From: Godfried Subject: TRICK question about systems of linear equations Thanks for your answer; you clarified a lot. I also found a geometric explanation, here it is: "If you have the graphs of 2 linear equations with 2 variables (x,y), there's almost always an intersection point, which is the unique simultaneous solution." \ / \/ /\ / \ Also, if you have equations in 3D space, you need 3 equations instead of 2. But, if 2 lines can intersect in 3D space (which I think is possible), what's the need, then, for a third line? Date: 10/30/2003 at 10:10:05 From: Doctor Douglas Subject: Re: TRICK question about systems of linear equations Hi again, Godfried. Excellent followup! In 3D, the relevant geometric objects are planes, not lines: Ax + By + Cz = D <---- a plane and it is not too hard to see that 2 planes that intersect in 3D space do so in a (1D) line, and that a third plane, if it intersects that line, will generate the point that is the unique solution to all three linear equations. In 4D, the relevant geometric objects are 3D-spaces. They mutually intersect in 2D-planes. Three of these spaces will in general specify a 1D-line and four of these spaces will in general determine the point that is the unique solution to all four linear equations. Tying the geometric interpretation into some ideas from my first reply, note that parallel lines in 2D relate to an inconsistent system: y = 2x + 3 y = 2x - 4 Similarly, two lines which are identical relate to an underconstrained system since a unique intersection point cannot be determined. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ Date: 10/30/2003 at 11:19:04 From: Godfried Subject: Thank you (TRICK question about systems of linear equations) Thanks for your clear explanation, and for preventing me drawing any more lines in 3D space. ;) |
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