Chapter 6, Problem 18CT

To determine

To determine if it is possible for the system to have infinitely many solutions if the graphs of the equations are not same.

Answer to Problem 18CT

No, it is not possible

Explanation of Solution

Given:

A system of equation:

  $a_1+b_1+c_1=0$ ….(i)

  $a_2+b_2+c_2=0$ ….(ii)

where $a_1$ , $b_1$ , $c_1$ , $a_2$ , $b_2$ , and $c_2$ are real numbers.

A system of equations has 3 forms:

1. Consistent system: The equations have only one solution and their graphs intersect at only one point (unique solution).
2. Dependent system: The equations have infinitely many solutions and their graphs are coinciding.
3. Inconsistent system: The equations have no solution and their graphs are parallel or non-intersecting.

It is not possible for the system to have infinitely many solutions if their graphs are not the same line.

Conclusion:

Therefore, for a system to have infinitely many solutions, the graphs must be coinciding.

Hence it is not possible for a system to have infinitely many solutions and not having same graph.