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Let there be a set of $3$ equations as follows :

\begin{align*} a_1x + b_1 y + c_1 z &= d_1 \\ a_2x + b_2 y + c_2 z &= d_2 \\ a_3x + b_3 y + c_3 z &= d_3 \end{align*}

Do we have a set of simple rules or conditions under which the above set of linear equations will have a unique solution, no solution or infinite solutions? I tried to find the information on this on the web but couldn't find any.

I have read about this back in my college and I know that we can determine the uniqueness of solution based on the relationship between the rank of the coefficient matrix, the rank of an augmented matrix and the number of variables in the equation but I am looking for some simple ratio relationships between the coefficients of variables as we have for a linear system of equations in 2 variables. Please help !!!

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  • $\begingroup$ What relation in two variables are you refering to? $\endgroup$ Jun 22, 2021 at 9:06
  • $\begingroup$ @Learn'dAstronomer : Like if $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ then we have unique solution for a system of linear equations in two variables. $\endgroup$
    – Ganit
    Jun 22, 2021 at 9:10
  • $\begingroup$ There are no ratio relationships, but look up Cramer's rule, that gives you what you need. The relationship is in matrix form. $\endgroup$ Jun 22, 2021 at 9:20

2 Answers 2

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Simplest way:

These equations are also Eq. of planes in 3D.

If any two panes are parallel then no solution.

If two of them are coincident.Then planes meet in a line and there are many solutions.

If all three planes are coincident, then many solutions.

Else, let $z=k$ and solve first two equations get $x,y$ in terms of $k$, put them in third equation. Three mutully exclusive things can happen.

1-$k$ gets determined, so unique solution: Planes meeting in a point.

2-$k$ disappears leaving a true statement like $3=3$, so many solutions possible for any real value of $k$: Planes meeting in a line.

3-$k$ disappears leaving a flase statement like $3=4$, so no solutions possible for any real value of $k$:Planes forming an open prism.

These situation can also be told in terms of Cramer determinants, adjoint or rank. of a matrix. When no solutions, system is called inconsistent. If unique or many solutions, the system is called consistent.

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It has a unique solution if and only if the determinant

$$\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{vmatrix}$$

is non-zero. This corresponds to the fact that the system

$$\begin{align*} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \\ \end{align*}$$

has a unique solution if and only if $a_1 b_2 - a_2b_1 \neq 0$ in two variables. This can be expressed as

$$\frac{a_1}{a_2} \neq \frac{b_1}{b_2},$$

which is the ratio condition you are referring to. This reflects the geometric fact that the system has a unique solution if and only if the lines represented by the equations are not parallel. In three dimensions, however, the equations correspond to planes, and they need not only be non-parallel, but they also need to be so that no straight line is contained in more than one of the three planes. I don't think you would be able to derive a simple ratio relationship in the same way since (1) the determinant has 6 terms as opposed to 2, so you cannot just divide yourself to a nice ratio as in the two-dimensional case and (2) that ratio condition would have to encode more than the one geometric condition you have in two dimensions.

As mentioned in the comments by Ritam_Dasgupts Cramer's rule is a nice matrix form of the solutions of $n \times n$ systems. It is however terribly inefficient computationally, so it is mostly of theoretical interest.

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