Question 201023


Start with the given system of equations:

{{{system(3r-5s=-3,5r+3s=29)}}}



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



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



{{{5(5r+3s)=5(29)}}} Multiply the both sides of the second equation by 5.



{{{25r+15s=145}}} Distribute and multiply.



So we have the new system of equations:

{{{system(9r-15s=-9,25r+15s=145)}}}



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-15s)+(25r+15s)=(-9)+(145)}}}



{{{(9r+25r)+(-15s+15s)=-9+145}}} Group like terms.



{{{34r+0s=136}}} Combine like terms.



{{{34r=136}}} Simplify.



{{{r=(136)/(34)}}} Divide both sides by {{{34}}} to isolate {{{r}}}.



{{{r=4}}} Reduce.



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



{{{9(4)-15s=-9}}} Plug in {{{r=4}}}.



{{{36-15s=-9}}} Multiply.



{{{-15s=-9-36}}} Subtract {{{36}}} from both sides.



{{{-15s=-45}}} Combine like terms on the right side.



{{{s=(-45)/(-15)}}} Divide both sides by {{{-15}}} to isolate {{{s}}}.



{{{s=3}}} Reduce.



So the solutions are {{{r=4}}} and {{{s=3}}}.



This means that the system is consistent and independent.