Question 303286


Start with the given system of equations:

{{{system(3x-2y=13,5x+7y=22)}}}



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



{{{21x-14y=91}}} Distribute and multiply.



{{{2(5x+7y)=2(22)}}} Multiply the both sides of the second equation by 2.



{{{10x+14y=44}}} Distribute and multiply.



So we have the new system of equations:

{{{system(21x-14y=91,10x+14y=44)}}}



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



{{{(21x-14y)+(10x+14y)=(91)+(44)}}}



{{{(21x+10x)+(-14y+14y)=91+44}}} Group like terms.



{{{31x+0y=135}}} Combine like terms.



{{{31x=135}}} Simplify.



{{{x=(135)/(31)}}} Divide both sides by {{{31}}} to isolate {{{x}}}.



------------------------------------------------------------------



{{{21x-14y=91}}} Now go back to the first equation.



{{{21(135/31)-14y=91}}} Plug in {{{x=135/31}}}.



{{{2835/31-14y=91}}} Multiply.



{{{31(2835/cross(31)-14y)=31(91)}}} Multiply both sides by the LCD {{{31}}} to clear any fractions.



{{{2835-434y=2821}}} Distribute and multiply.



{{{-434y=2821-2835}}} Subtract {{{2835}}} from both sides.



{{{-434y=-14}}} Combine like terms on the right side.



{{{y=(-14)/(-434)}}} Divide both sides by {{{-434}}} to isolate {{{y}}}.



{{{y=1/31}}} Reduce.



So the solutions are {{{x=135/31}}} and {{{y=1/31}}}.



Which form the ordered pair *[Tex \LARGE \left(\frac{135}{31},\frac{1}{31}\right)].



This means that the system is consistent and independent.