Question

Is the following statement correct? Explain.

A system of two equations has no solution if the graphs of the two equations are coincident lines.

Answer 1

Answer: The answer is NO.

Explanation: The given statement is -

If the graph of two equations are coincident lines, then that system of equations will have no solution.

We are to check whether the above statement is correct or not.

Any two equations having graphs as coincident lines are of the form -

`ax+by=c,\\\\dax+dby=dc,\\\\\textup{where}~~d\\eq 1.`

If we take d = 1, then both the equations will be same.

Now, subtracting the second equation from first, we have

`a(1-d)x+b(1-d)y=c(1-d)\\\\\Rightarrow ax+by=c,~\textup{since}~d\\eq 1,~\textup{so}~1-d\\eq 0.`

Again, we will get the first equation, which is linear in two unknown variables. So, the system will have infinite number of solutions, which consists of the points lying on the line.

For example, see the attached figure, the graphs of following two equations is drawn and they are coincident. Also, the result is again the same straight line which has infinite number of points on it. These points makes the solution for the following system.

`2x+5y=10,\\\\6x+15y=30.`

Thus, the given statement is not correct.