Lesson Equations with up to 2 solutions

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A system of 2 equations can have 2 solutions, for x and y,no solutions, or infinite solutions. Example,

{{{system(y=4x-5,4x-y=8)}}} Substitute 4x-5 for y,

{{{4x-(4x-5)=8}}}

{{{4x-4x+5=8}}}

{{{5=8}}} This is a false statement so there are no solutions. This graph shows,

{{{graph(450,450,-10,10,-10,10,y=4x-5,y=8+4x)}}} As you can see, the two lines are parallel to each other so they will never meet so there is no solution.

{{{system(4x+6y=16,5x-2y=1)}}} Multiply the top by 5 and the bottom by 4,

{{{system(20x+30y=80,20x-8y=4)}}} Subtract the 2 equations,

{{{38y=76}}} {{{highlight(y=2)}}} Substitute 2 for y,

{{{4x+12=16}}} {{{4x=4}}} {{{highlight(x=1)}}} The lines intersect at the point (1,2) in the following graph,

{{{graph(450,450,-10,10,-10,10,y=8/3-(2/3)x,y=(5/2)x-1/2)}}}

{{{system(y=3x,6x-2y=0)}}} Substitute 3x for y,

{{{6x-6x=0}}}

{{{0=0}}} Infinite amount of solutions.

{{{graph(450,450,-10,10,-10,10,y=3x,y=3x)}}} As you can see, the 2 lines intersect.  To learn more about graphs, check out this <A HREF=/my/graphs-and-inequalities.lesson?content_action=show_dev>lesson</A>.