Question 299194


Start with the given system of equations:

{{{system(3r-4s=5,4r+3s=15)}}}



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



{{{9r-12s=15}}} Distribute and multiply.



{{{4(4r+3s)=4(15)}}} Multiply the both sides of the second equation by 4.



{{{16r+12s=60}}} Distribute and multiply.



So we have the new system of equations:

{{{system(9r-12s=15,16r+12s=60)}}}



Why did we just do all that? You'll see below:



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



{{{(9r-12s)+(16r+12s)=(15)+(60)}}}



{{{(9r+16r)+(-12s+12s)=15+60}}} Group like terms.



{{{25r+0s=75}}} Combine like terms.



Notice how the 's' terms add to 0 and cancel out. So we're now left with a simple equation with one unknown. 



{{{25r=75}}} Simplify.



{{{r=(75)/(25)}}} Divide both sides by {{{25}}} to isolate {{{r}}}.



{{{r=3}}} Reduce.



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{{{9r-12s=15}}} Now go back to the first equation.



{{{9(3)-12s=15}}} Plug in {{{r=3}}}.



{{{27-12s=15}}} Multiply.



{{{-12s=15-27}}} Subtract {{{27}}} from both sides.



{{{-12s=-12}}} Combine like terms on the right side.



{{{s=(-12)/(-12)}}} Divide both sides by {{{-12}}} to isolate {{{s}}}.



{{{s=1}}} Reduce.



So the solutions are {{{r=3}}} and {{{s=1}}}.



Which form the ordered pair *[Tex \LARGE \left(3,1\right)].



This means that the system is consistent and independent.