1st case: the system has one solution
For example:
Given this system:
[tex]\begin{gathered} x+y=5 \\ 2x-y=4 \\ \text{Let:} \\ x+y=5\text{ (1)} \\ 2x-y=4\text{ (2)} \\ \text{ Using elimination method:} \\ (1)+(2)\colon \\ x+2x+y-y=9 \\ 3x=9 \\ x=\frac{9}{3} \\ x=3 \\ y=5-x \\ y=5-3 \\ y=2 \end{gathered}[/tex]
graphically, a system has a solution if the two lines intersect, the point of intersection is the solution.
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2nd case: the system has no solution
A system has no solution, when the lines are parallel and have different intercepts, for example:
[tex]\begin{gathered} y=2x+1 \\ y=2x-3 \end{gathered}[/tex]
as you can see the lines never cross each other.
3rd case: the system has infinitely many solutions
occurs when one line is a scalar multiple of the other, in other words it is the same line. for example:
[tex]\begin{gathered} x+y=5 \\ 2x+2y=10 \end{gathered}[/tex]
