Infinite solutions to a system means there are infinite set of values for its variables for which the system is true. The needed equation is: [tex]2x+ 2y = 4[/tex]
When does a system of linear equation have infinite solutions?
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.
Since the first line(linear equations are equation of lines) given is [tex]x+y = 2[/tex]
For getting infinite solutions, we need another line which would be coincident on that same line. Since the second line is coincident(lying exactly over) to the first line, thus, its equation is same as that of the first line.
Thus, second line's equation is x + y = 2
Option second is correct here since we have:
[tex]2x + 2y = 4\\\\\text{dividing by 2}\\\\x + y = 2[/tex]
So both are equivalent equations(those equations who might look different but are same in their simplified forms).
Thus, The needed equation is: [tex]2x+ 2y = 4[/tex] for which the system
[tex]x+y = 2\\2x + 2y = 4[/tex] will have infinitely many solutions.
Learn more about solutions of system of linear equations here:
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