Hello,

I am able to solve Systems of Linear Equation in Three Variables. But when I tried to search Linear Equation in Three Variables in Google, it always end up being Systems of Linear Equation.

For example, we have \(3x+2y+4z\). Is it possible to obtain all possible ordered triplets that will make the equation true, just like what we do in Linear Equation in Two Variables?

Example \(2x+3y=12\).

These one the solutions to this equation.

\((3,2)\).

I also am aware that this involves three variables so we can't just make it like a \(y=mx+b\) where y depends solely on x.

I know the answer, or probably one of the answers, is \(x=2, y=3, z=5\).

So how to solve "Linear Equation in Three Variables", not "Systems of Linear Equations in Three Variables"?

Thank you very much.

 

Not sure what you meant. Are you talking about "Linear Diophantine Equations with three variables"?

ax + by + cz = d
(x, y, z are variables and a, b, c, d are integer coefficients)
If so, take a look at this:
http://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_5_print.pdf

 

It could be non-integer. I just want to know how to obtain all the possible ordered triplets that will make the equation true.

 

What equation?

You've given 3x+2y+4z. This is an expression, not an equation. What does it mean for this to be "true"?

If you have 3x+2y+4z=C, then consider what happens if you solve for, say, y:

y = (C-3x-4z)/2

Since you aren't requiring solutions to be integers, there will be an infinite number of solutions. Pick an x and pick a z and I will tell you the corresponding value of y. The same is true with an equation in two variables. So the notion of finding all of the ordered pairs or triplets is rather nonsensical.

 

But when I tried to search Linear Equation in Three Variables in Google, it always end up being Systems of Linear Equation.



For example, we have . Is it possible to obtain all possible ordered triplets that will make the equation true, just like what we do in Linear Equation in Two Variables?

Yes it is possible to obtain the triplets you ask (although as WBahn has pointed out there are an infinite number, but is not nonsensical. It is just not used in electrical engineering. It has many uses in other areas, particularly chemical engineering and economics.)

The part of mathematics you are seeking is called

Linear Programming

(Note 'programming' in this has nothing to do with computers although they can be used for the solution)

Not linear equations.

There are lots of good online references. The standard method of resolution is called the simplex method, which you can also search for.

 

What equation?

You've given 3x+2y+4z. This is an expression, not an equation. What does it mean for this to be "true"?

If you have 3x+2y+4z=C, then consider what happens if you solve for, say, y:

y = (C-3x-4z)/2

Since you aren't requiring solutions to be integers, there will be an infinite number of solutions. Pick an x and pick a z and I will tell you the corresponding value of y. The same is true with an equation in two variables. So the notion of finding all of the ordered pairs or triplets is rather nonsensical.

Geometrically, equations like the one above represent a plane. Given two such equations they either have no points in common, or they intersect in a line. Given three such equations the situation is more complicated and includes no points of intersection, a single point of intersection, and one or more lines of intersection.

http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx

 

Yes it is possible to obtain the triplets you ask (although as WBahn has pointed out there are an infinite number, but is not nonsensical. It is just not used in electrical engineering. It has many uses in other areas, particularly chemical engineering and economics.)

The part of mathematics you are seeking is called

Linear Programming

(Note 'programming' in this has nothing to do with computers although they can be used for the solution)

Not linear equations.

There are lots of good online references. The standard method of resolution is called the simplex method, which you can also search for.

By "nonsensical" I only meant that the notion of writing out a list of all the ordered triplets doesn't make sense. The equation itself defines the conditions that x, y, and z have to meet. If you want to write them as a triplet, then you just have <x,y,g(x,y)> where g(x,y) is simply f(x,y,z) solved for z. Pick any x and y and as along as g(x,y) is finite, you have a triplet.