'c' can be any real number except 4 for which the considered system would have no solution.
When does a system of linear equation have no solution?
When both the lines(represented by both the equations) are parallel and not coincident, then that system of linear equations have no solutions as there is no common point on both line.
If they are coincident (lying over each other), then there will have infinite solution since in that case, they will have infinite points in common.
Thus, if two lines are not in any of above case, they intersect at single point and in that case, the considered system of equation has unique single solution.
How to get the slope intercept form of a straight line equation?
If the slope of a line is m and the y-intercept is c, then the equation of that straight line is given as:
[tex]y = mx +c[/tex]
Parallel lines(non-coincident) have same slope but different y-intercept.
Converting these equations to slope-intercept form, we get:
For first line:
[tex]\dfrac{1}{2}x + \dfrac{1}{5}y = 2\\\\ \dfrac{1}{5}y = 2 - \dfrac{1}{2}x\\\\y = 10 - \dfrac{5}{2}x\\\\y = -\dfrac{5}{2}x + 2[/tex]
and
For second line:
[tex]5x + 2y = c\\\\y = \dfrac{1}{2}c - \dfrac{5}{2}x\\\\y = -\dfrac{5}{2}x + \dfrac{c}{2}[/tex]
For no-solution, we need both lines to stay parallel but not coincident so that there is no common point (solution of this system) between those two lines.
Thus, we get:
[tex]2 \neq \dfrac{c}{2}\\\\\text{Multiplying 2 on both the sides}\\\\c \neq 4[/tex]
Since 'y' assumes real numbers assumingly, so 'c' can be any real number except 4 for which the considered system would have no solution.
Below is image attached of those parallel lines when c = 10.
Thus, 'c' can be any real number except 4 for which the considered system would have no solution.
Learn more about the case of 'no solution' here:
https://brainly.com/question/26254258
