Question 175701

Start with the given system of equations:

{{{system(3r-4s=0,2r+5s=23)}}}



{{{5(3r-4s)=5(0)}}} Multiply the both sides of the first equation by 5.



{{{15r-20s=0}}} Distribute and multiply.



{{{4(2r+5s)=4(23)}}} Multiply the both sides of the second equation by 4.



{{{8r+20s=92}}} Distribute and multiply.



So we have the new system of equations:

{{{system(15r-20s=0,8r+20s=92)}}}



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



{{{(15r-20s)+(8r+20s)=(0)+(92)}}}



{{{(15r+8r)+(-20s+20s)=0+92}}} Group like terms.



{{{23r+0s=92}}} Combine like terms.



{{{23r=92}}} Simplify.



{{{r=(92)/(23)}}} Divide both sides by {{{23}}} to isolate {{{r}}}.



{{{r=4}}} Reduce.



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{{{15r-20s=0}}} Now go back to the first equation.



{{{15(4)-20s=0}}} Plug in {{{r=4}}}.



{{{60-20s=0}}} Multiply.



{{{-20s=0-60}}} Subtract {{{60}}} from both sides.



{{{-20s=-60}}} Combine like terms on the right side.



{{{s=(-60)/(-20)}}} Divide both sides by {{{-20}}} to isolate {{{s}}}.



{{{s=3}}} Reduce.



So our answer is {{{r=4}}} and {{{s=3}}}.



This means that the system is consistent and independent.