Question 183045


Start with the given system of equations:

{{{system(5x-4y=2,4x-5y=-1)}}}



{{{4(5x-4y)=4(2)}}} Multiply the both sides of the first equation by 4.



{{{20x-16y=8}}} Distribute and multiply.



{{{-5(4x-5y)=-5(-1)}}} Multiply the both sides of the second equation by -5.



{{{-20x+25y=5}}} Distribute and multiply.



So we have the new system of equations:

{{{system(20x-16y=8,-20x+25y=5)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(20x-16y)+(-20x+25y)=(8)+(5)}}}



{{{(20x+-20x)+(-16y+25y)=8+5}}} Group like terms.



{{{0x+9y=13}}} Combine like terms.



{{{9y=13}}} Simplify.



{{{y=(13)/(9)}}} Divide both sides by {{{9}}} to isolate {{{y}}}.



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{{{20x-16y=8}}} Now go back to the first equation.



{{{20x-16(13/9)=8}}} Plug in {{{y=13/9}}}.



{{{20x-208/9=8}}} Multiply.



{{{9(20x)-cross(9)(208/cross(9))=9(8)}}} Multiply EVERY term by the LCD {{{9}}} to clear any fractions.



{{{180x-208=72}}} Distribute and multiply.



{{{180x=72+208}}} Add {{{208}}} to both sides.



{{{180x=280}}} Combine like terms on the right side.



{{{x=(280)/(180)}}} Divide both sides by {{{180}}} to isolate {{{x}}}.



{{{x=14/9}}} Reduce.



So the solutions are {{{x=14/9}}} and {{{y=13/9}}}.



Which form the ordered pair *[Tex \LARGE \left(\frac{14}{9},\frac{13}{9}\right)].



This means that the system is consistent and independent.