When does a system of linear equations have a unique solution?

Question

What is a systematic way of determining whether a system of equations has a unique solution?

For example, this system of $2$ equations with $3$ unknowns: $$2a+b+c=0\\ 6a+2b+c=0.$$

Answer 1

With only those two equations, there is not a unique solution for $a$, $b$, and $c$.

Subtracting the first equation from the second yields $4a+b=0$ or $b=-4a$.

Substituting $-4a$ for $b$ in either equation yields $-2a+c=0$ or $c=2a$.

Can you verify that, if $a$ is any real number,

then with $b=-4a$ and $c=2a$ both equations are satisfied?

Answer 2

1. Given a system of of $q$ independent linear equations with $n$ variables:

   • if $q>n,$ then the system is inconsistent, i.e., has no solution;
   • if $q=n,$ then it is inconsistent or has a unique solution;
   • if $q<n,$ then it is inconsistent or has infinitely many solutions.

2. It is easy to verify that your given system, which has only two equations, is independent; then, since it has $q=2\ne3=n,$ it has no unique solution.

3. On the other hand, a system of dependent linear equations can have either no solution or a unique solution or infinitely many solutions. For example, this system has a unique solution (notice that here, $q>n$): $$x-y=0\\x+y=6\\7x+3y=30.$$

4. A linear system with $n$ variables has a unique solution iff the rank of its coefficient matrix and its augmented matrix both equal $n.$

Answer 3

Like any system of equations, you can subtract the equation on the bottom from the one on the top to remove a variable from both equations, so long as one variable on the bottom equals one on the top. In this case, that variable is c. After subtracting, you will be left with just -4a-b=0 or, more cleanly, 4a+b=0. It is possible to formulate this in terms of a linear equation b=-4a. So no, there is no one answer. There is one value of b corresponding to each value of a.

Answer 4

You can imagine your equations as an equations of planes in 3d (where a is x, b is y and c is z). If you'd had 3 linear equations - 3 planes and not parallel to each-other, you would get an intersection - one point => single soultion.

When you have two planes, they give an intersection as a line. Basically, this line in 3d space is your solution.

Look here for visualization: https://www.math3d.org/xIOmOlVN