Question

What defines an inconsistent linear system? Can you solve an inconsistent linear system?

Answer 1

Hint: In this problem we need to define an inconsistent linear system. First, we will see what is a linear system of equations, after that we will come to what is an inconsistent linear system. Once we deal with the definition of the inconsistent linear system, we will see some examples for the inconsistent linear system and try to solve them.

Complete step-by-step answer:
We know that the equation of a line in any coordinate plane is considered as the linear equation. For example, the linear equation $x+2y=9$ represents a line in the $xy$ plane while $x+z=3$ represents a line in the $xz$ plane.
Now the system of linear equations means when two or more number of linear equations work together or forms a relation between both of them. A system of linear equations should have a minimum of two linear equations. For example,
$\begin{align}
  & x+y=6 \\
 & -3x+y=2 \\
\end{align}$ is a system of linear equations in the $xy$ plane.
Depending upon the number of solutions for a linear system of equations, the linear system of equations are classified into three types. Inconsistent systems of linear equations are also one type of system of linear equations which has no solution that means the linear equations which form an inconsistent system of linear equations don’t have any relation between them. The following are some examples of inconsistent systems of linear equations.
$\begin{align}
  & x+2y=3 \\
 & 3x+6y=5 \\
\end{align}$, $\begin{align}
  & x-y=8 \\
 & 5x-5y=25 \\
\end{align}$.
When we try to solve the above equations, we won’t get any identity. For example, consider the system $\begin{align}
  & x+2y=3 \\
 & 3x+6y=5 \\
\end{align}$.
In the above system multiply the equation $x+2y=3$ with $3$ and subtract it from $3x+6y=5$, then we will get
$\begin{align}
  & 3x+6y-3\left( x+2y \right)=5-3\left( 3 \right) \\
 & \Rightarrow 3x+6y-3x-6y=5-9 \\
 & \Rightarrow 0=-4 \\
\end{align}$
Here we got $0=-4$ which is not a math identity, so the solution for the system does not exist.

Note: In this problem we have only discussed the system of linear equations in two variables only that means $x$ and $y$ only. So many systems of linear equations are there with three or more than three variables also. The concept of the system of linear equations is used in solving distance-time-rate problems, economics and geometry problems.