Question

How does a system of linear equations have no solution\[?\]

Answer 1

Hint: Here in this question, we explain how a system of linear equations will be in the form of \[y = {m_1}x + a\] and \[y = {m_2}x + b\]have no solution. If \[{m_1} = {m_2}\] and \[a \ne b\] then there will be no solution. This type of equation is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel.

Complete step-by-step solution:
An equation of the form \[y = mx + c\]is a linear equation . While considering the system of linear equations, we can find the number of solutions by comparing the coefficients of the equations. Also, we can find whether the system of equations has no solution or infinitely many solutions by graphical method.
Let us consider the pair of linear equations
\[y = {m_1}x + a\]
\[y = {m_2}x + b\]
Here \[a\], \[b\], \[{m_1}\], \[{m_2}\] are real numbers. where m is slope.
If \[{m_1} = {m_2}\] and \[a \ne b\] then there will be no solution. This type of equation is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel.
Consider an example to check the linear equations have no solution
\[y = 4x + 3\]
\[y = 4x + 5\]
Here these systems of linear equations satisfy the condition \[{m_1} = {m_2}\] and \[a \ne b\], both the lines have the same slope 4 which means lines that are parallel never intersect. That being said, there is no solution to this problem.

Note: While solving the system of equations sometimes we get the value of an unknown variable. By the equation we can plot the graph also, if they are parallel to each other then we will not be able to find the solution for the system of equations. Hence the above example represents the system of equations that has no solutions.