Question 163048


Start with the given system of equations:

{{{system(3x+5y=4,5x+2y=9)}}}



{{{5(3x+5y)=5(4)}}} Multiply the both sides of the first equation by 5.



{{{15x+25y=20}}} Distribute and multiply.



{{{-3(5x+2y)=-3(9)}}} Multiply the both sides of the second equation by -3.



{{{-15x-6y=-27}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15x+25y=20,-15x-6y=-27)}}}



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



{{{(15x+25y)+(-15x-6y)=(20)+(-27)}}}



{{{(15x+-15x)+(25y+-6y)=20+-27}}} Group like terms.



{{{0x+19y=-7}}} Combine like terms. Notice how the x terms cancel out.



{{{19y=-7}}} Simplify.



{{{y=(-7)/(19)}}} Divide both sides by {{{19}}} to isolate {{{y}}}.



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{{{15x+25y=20}}} Now go back to the first equation.



{{{15x+25(-7/19)=20}}} Plug in {{{y=-7/19}}}.



{{{15x-175/19=20}}} Multiply.



{{{19(15x-175/cross(19))=19(20)}}} Multiply both sides by the LCD {{{19}}} to clear any fractions.



{{{285x-175=380}}} Distribute and multiply.



{{{285x=380+175}}} Add {{{175}}} to both sides.



{{{285x=555}}} Combine like terms on the right side.



{{{x=(555)/(285)}}} Divide both sides by {{{285}}} to isolate {{{x}}}.



{{{x=37/19}}} Reduce.



So our answer is {{{x=37/19}}} and {{{y=-7/19}}}.



Which form the ordered pair *[Tex \LARGE \left(\frac{37}{19},\frac{-7}{19}\right)].



This means that the system is consistent and independent.