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if the equation winds up with an equality and no variables, then you are dealing with an infinite number of solutions.
\n" ); document.write( "example:
\n" ); document.write( "3 = 3
\n" ); document.write( "0 = 0
\n" ); document.write( "etc.
\n" ); document.write( "if the equation winds up with no equality and no variables, then you are dealing with no solutions.
\n" ); document.write( "example:
\n" ); document.write( "2 = 3
\n" ); document.write( "0 = 5
\n" ); document.write( "etc.
\n" ); document.write( "an example of a system of equations with infinite number of solutions.
\n" ); document.write( "x + y = 2
\n" ); document.write( "2x + 2y = 4
\n" ); document.write( "you solve this system of equations by multiplying the first equation by 2 to get:
\n" ); document.write( "2x + 2y = 4 (first equation multiplied by 2)
\n" ); document.write( "2x + 2y = 4 (second equation)
\n" ); document.write( "when you subtract the first equation from the second equation, you get:
\n" ); document.write( "0 + 0 = 0 which becomes 0 = 0
\n" ); document.write( "this indicates an infinite number of solutions.
\n" ); document.write( "any value for x and any value for y that satisfies one of the equations will automatically satisfy the other equation.
\n" ); document.write( "for example:
\n" ); document.write( "if x = 5 and y = -3, then x + y = 2 becomes 5 - 3 = 2 which becomes 2 = 2 which is good.
\n" ); document.write( "plugging those same values into the second equation gets:
\n" ); document.write( "2x + 2y = 4 becomes 2*5 - 2*3 = 4 which becomes 10 - 6 = 4 which becomes 4 = 4 which is good.
\n" ); document.write( "any combination of x and y that satisfies one of the equation will satisfy the other.
\n" ); document.write( "an example of no solutions is as follows:
\n" ); document.write( "x + y = 2
\n" ); document.write( "2x + 2y = 7
\n" ); document.write( "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:
\n" ); document.write( "2x + 2y = 4 (first equation multiplied by 2)
\n" ); document.write( "2x + 2y = 7 (second equation)
\n" ); document.write( "when you subtract the first equation from the second equation, you get:
\n" ); document.write( "0 + 0 = 3 which becomes 0 = 3.
\n" ); document.write( "this is false, so there is no solution to this system of equations.
\n" ); document.write( "we can graph both the infinite number of solutions and the no solution to show you how the graph will look.
\n" ); document.write( "your first 2 equations were:
\n" ); document.write( "x + y = 2
\n" ); document.write( "2x + 2y = 4
\n" ); document.write( "solve for y in both equations and you will get:
\n" ); document.write( "y = -x + 2
\n" ); document.write( "y = -x + 2
\n" ); document.write( "these equations are identical and so their graphs will coincide and look like the same line.
\n" ); document.write( "your second 2 equations were:
\n" ); document.write( "x + y = 2
\n" ); document.write( "2x + 2y = 7
\n" ); document.write( "solve for y in both equations and you will get:
\n" ); document.write( "y = -x + 2
\n" ); document.write( "y = -x + 7/2
\n" ); document.write( "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.
\n" ); document.write( "note that all equations are in the slope intercept form.
\n" ); document.write( "that form is y = mx + b
\n" ); document.write( "m is the slope and b is the y intercept.
\n" ); document.write( "if the slopes are the same and the y intercepts are different then the lines are parallel and will never intersect.
\n" ); document.write( "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.
\n" ); document.write( "the graph of the first 2 equations where we had an infinite number of solutions is shown below:
\n" ); document.write( "
\n" ); document.write( "the graph of the second 2 equations where we had no solution is shown below:
\n" ); document.write( "
\n" ); document.write( "in the first graph, the 2 lines are superimposed on each other because the equations are identical so it looks like you have one line, but you really have 2.
\n" ); document.write( "the only way to know that is to remove one of the equations from the graph and then you will see that the graph is still there.\r
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