Step-by-step explanation:
Note that when lines intersect, that when they show a solution.
All Solutions:
When we graph two equations, and they line on the same curve thus all points will be a point of intersection..
Let say we have two linear equations.
[tex]4x + 4y = 16[/tex]
[tex]x + y = 4[/tex]
This are all solutions, because
if we do the graphical way, when we graph this on a calculator or desmos, they will lie on the same curve so it has infinite points of intersections.
If we do algebraic way,
Notice that the top equation is the same as The bottom equation but just multiplied by 4. So the top and bottom equations are the same.
So whenever the two or more equations have same slope and same y intercept, the solutions will be all real solutions or infinite solutions.
One Solution:
In plane geometry, note that a line that isn't parallel to another one, will intersect eventually once.
This means that for example
[tex]4x + 4y = 16[/tex]
[tex]x + 2y = 32[/tex]
These equations have different slope so they will intersect once. Which means it'll have one solution.
NO Solutions:
Parallel Lines never intersect which means they have no solutions.
Consider this
[tex]4x + 4y = 16[/tex]
[tex]x + y = 1[/tex]
We know the slopes are the same but y intercept is different they will never intersect.
