Question 388335

Start with the given system of equations:

{{{system(2r-5s=-33,5r+2s=48)}}}



{{{2(2r-5s)=2(-33)}}} Multiply the both sides of the first equation by 2.



{{{4r-10s=-66}}} Distribute and multiply.



{{{5(5r+2s)=5(48)}}} Multiply the both sides of the second equation by 5.



{{{25r+10s=240}}} Distribute and multiply.



So we have the new system of equations:

{{{system(4r-10s=-66,25r+10s=240)}}}



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



{{{(4r-10s)+(25r+10s)=(-66)+(240)}}}



{{{(4r+25r)+(-10s+10s)=-66+240}}} Group like terms.



{{{29r+0s=174}}} Combine like terms.



{{{29r=174}}} Simplify.



{{{r=(174)/(29)}}} Divide both sides by {{{29}}} to isolate {{{r}}}.



{{{r=6}}} Reduce.



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{{{4r-10s=-66}}} Now go back to the first equation.



{{{4(6)-10s=-66}}} Plug in {{{r=6}}}.



{{{24-10s=-66}}} Multiply.



{{{-10s=-66-24}}} Subtract {{{24}}} from both sides.



{{{-10s=-90}}} Combine like terms on the right side.



{{{s=(-90)/(-10)}}} Divide both sides by {{{-10}}} to isolate {{{s}}}.



{{{s=9}}} Reduce.



So the solutions are {{{r=6}}} and {{{s=9}}}.



This means that the system is consistent and independent.



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