Question 198899


Start with the given system of equations:

{{{system(5r-6s=21,6r+5s=74)}}}



{{{6(5r-6s)=6(21)}}} Multiply the both sides of the first equation by 6.



{{{30r-36s=126}}} Distribute and multiply.



{{{-5(6r+5s)=-5(74)}}} Multiply the both sides of the second equation by -5.



{{{-30r-25s=-370}}} Distribute and multiply.



So we have the new system of equations:

{{{system(30r-36s=126,-30r-25s=-370)}}}



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



{{{(30r-36s)+(-30r-25s)=(126)+(-370)}}}



{{{(30r+-30r)+(-36s+-25s)=126+-370}}} Group like terms.



{{{0r+-61s=-244}}} Combine like terms.



{{{-61s=-244}}} Simplify.



{{{s=(-244)/(-61)}}} Divide both sides by {{{-61}}} to isolate {{{s}}}.



{{{s=4}}} Reduce.



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{{{30r-36s=126}}} Now go back to the first equation.



{{{30r-36(4)=126}}} Plug in {{{s=4}}}.



{{{30r-144=126}}} Multiply.



{{{30r=126+144}}} Add {{{144}}} to both sides.



{{{30r=270}}} Combine like terms on the right side.



{{{r=(270)/(30)}}} Divide both sides by {{{30}}} to isolate {{{r}}}.



{{{r=9}}} Reduce.



So the solutions are {{{r=9}}} and {{{s=4}}}.



This means that the system is consistent and independent.