Question 221989


Start with the given system of equations:

{{{system(7r-5s=-12,5r-7s=76)}}}



{{{7(7r-5s)=7(-12)}}} Multiply the both sides of the first equation by 7.



{{{49r-35s=-84}}} Distribute and multiply.



{{{-5(5r-7s)=-5(76)}}} Multiply the both sides of the second equation by -5.



{{{-25r+35s=-380}}} Distribute and multiply.



So we have the new system of equations:

{{{system(49r-35s=-84,-25r+35s=-380)}}}



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



{{{(49r-35s)+(-25r+35s)=(-84)+(-380)}}}



{{{(49r+-25r)+(-35s+35s)=-84+-380}}} Group like terms.



{{{24r+0s=-464}}} Combine like terms. Notice how the y terms cancel out.



{{{24r=-464}}} Simplify.



{{{r=(-464)/(24)}}} Divide both sides by {{{24}}} to isolate {{{r}}}.



{{{r=-58/3}}} Reduce.



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{{{49r-35s=-84}}} Now go back to the first equation.



{{{49(-58/3)-35s=-84}}} Plug in {{{r=-58/3}}}.



{{{-2842/3-35s=-84}}} Multiply.



{{{3(-2842/cross(3)-35s)=3(-84)}}} Multiply both sides by the LCD {{{3}}} to clear any fractions.



{{{-2842-105s=-252}}} Distribute and multiply.



{{{-105s=-252+2842}}} Add {{{2842}}} to both sides.



{{{-105s=2590}}} Combine like terms on the right side.



{{{s=(2590)/(-105)}}} Divide both sides by {{{-105}}} to isolate {{{s}}}.



{{{s=-74/3}}} Reduce.



So the solutions are {{{r=-58/3}}} and {{{s=-74/3}}}