How can you tell if its no solution, one solution or infinite

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Question 467081: 1. how do you know when an equation has an infinite number of solutions - show an example.
2. how do you know when an equation has no solution - show an example.
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Answer by Theo(7272) About Me (Show Source):
You can put this solution on YOUR website!
if the equation winds up with an equality and no variables, then you are dealing with an infinite number of solutions.
example:
3 = 3
0 = 0
etc.
if the equation winds up with no equality and no variables, then you are dealing with no solutions.
example:
2 = 3
0 = 5
etc.
an example of a system of equations with infinite number of solutions.
x + y = 2
2x + 2y = 4
you solve this system of equations by multiplying the first equation by 2 to get:
2x + 2y = 4 (first equation multiplied by 2)
2x + 2y = 4 (second equation)
when you subtract the first equation from the second equation, you get:
0 + 0 = 0 which becomes 0 = 0
this indicates an infinite number of solutions.
any value for x and any value for y that satisfies one of the equations will automatically satisfy the other equation.
for example:
if x = 5 and y = -3, then x + y = 2 becomes 5 - 3 = 2 which becomes 2 = 2 which is good.
plugging those same values into the second equation gets:
2x + 2y = 4 becomes 2*5 - 2*3 = 4 which becomes 10 - 6 = 4 which becomes 4 = 4 which is good.
any combination of x and y that satisfies one of the equation will satisfy the other.
an example of no solutions is as follows:
x + y = 2
2x + 2y = 7
when you multiply the first equation by 2 to eliminate one of the variables, you wind up eliminating all of the variables and you get:
2x + 2y = 4 (first equation multiplied by 2)

2x + 2y = 7 (second equation)
when you subtract the first equation from the second equation, you get:
0 + 0 = 3 which becomes 0 = 3.
this is false, so there is no solution to this system of equations.
we can graph both the infinite number of solutions and the no solution to show you how the graph will look.
your first 2 equations were:
x + y = 2
2x + 2y = 4
solve for y in both equations and you will get:
y = -x + 2
y = -x + 2
these equations are identical and so their graphs will coincide and look like the same line.
your second 2 equations were:
x + y = 2
2x + 2y = 7
solve for y in both equations and you will get:
y = -x + 2
y = -x + 7/2
these equations have the same slope but have a different y intercept so they are parallel to each other. this means they will never intersect which means you have no common solution.
note that all equations are in the slope intercept form.
that form is y = mx + b
m is the slope and b is the y intercept.
if the slopes are the same and the y intercepts are different then the lines are parallel and will never intersect.
if the slopes are the same and the y intercepts are the same, then the lines are identical and you have an infinite number of solutions.
the graph of the first 2 equations where we had an infinite number

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