How will I know when an equation has infinitely many solutions?
when the equations are equal to eachother.
I would I explain that in 30 words?
For the linear equation, \[y =mx+b\] , you have parallel lines if the slopes, m, are the same and the y-intercepts, b, are different. These parallel lines will lie on top of each other, creating an intersection for every value of x, when the slopes and y-intercepts are the same.
Ok, I understand that. now the next part that I do not understand and know how to explain is when the equation has no solution?
When the lines are parallel they will never touch. They will be spaced from eachother by a set distance from -infinity to infinity.
I am not sure that I understand what that means? can you clarify?
Imagine two lines with the same slope. Imagine that one of them crosses the y-axis at 1, and the other at 2. Although these lines have the same slope and point in the same direction, they will never touch. They will never Intersect each other. You get solutions when the graphs of your functions intersect. These intersections are solutions because the coordinates for each intersection are values that exist for each function simultaneously.
Here, look at this. http://www.wolframalpha.com/input/?i=graph+y%3D2x%2B2+and+y%3D2x%2B1
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