Question 554869

Start with the given system of equations:

{{{system(5x+3y=-11,7x-2y=13)}}}



{{{2(5x+3y)=2(-11)}}} Multiply the both sides of the first equation by 2.



{{{10x+6y=-22}}} Distribute and multiply.



{{{3(7x-2y)=3(13)}}} Multiply the both sides of the second equation by 3.



{{{21x-6y=39}}} Distribute and multiply.



So we have the new system of equations:

{{{system(10x+6y=-22,21x-6y=39)}}}



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



{{{(10x+6y)+(21x-6y)=(-22)+(39)}}}



{{{(10x+21x)+(6y+-6y)=-22+39}}} Group like terms.



{{{31x+0y=17}}} Combine like terms.



{{{31x=17}}} Simplify.



{{{x=(17)/(31)}}} Divide both sides by {{{31}}} to isolate {{{x}}}.



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{{{10x+6y=-22}}} Now go back to the first equation.



{{{10(17/31)+6y=-22}}} Plug in {{{x=17/31}}}.



{{{170/31+6y=-22}}} Multiply.



{{{31(170/cross(31)+6y)=31(-22)}}} Multiply both sides by the LCD {{{31}}} to clear any fractions.



{{{170+186y=-682}}} Distribute and multiply.



{{{186y=-682-170}}} Subtract {{{170}}} from both sides.



{{{186y=-852}}} Combine like terms on the right side.



{{{y=(-852)/(186)}}} Divide both sides by {{{186}}} to isolate {{{y}}}.



{{{y=-142/31}}} Reduce.



So the solutions are {{{x=17/31}}} and {{{y=-142/31}}}.



Which form the ordered pair *[Tex \LARGE \left(\frac{17}{31},-\frac{142}{31}\right)].



This means that the system is consistent and independent.



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