Question 202045
"the difference between two numbers is 11" means that {{{x-y=11}}}



"the larger number is added to two times the smaller the sum is 59" tells us that {{{2x+y=59}}}





Start with the given system of equations:

{{{system(x-y=11,x+2y=59)}}}



{{{2(x-y)=2(11)}}} Multiply the both sides of the first equation by 2.



{{{2x-2y=22}}} Distribute and multiply.



So we have the new system of equations:

{{{system(2x-2y=22,x+2y=59)}}}



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



{{{(2x-2y)+(x+2y)=(22)+(59)}}}



{{{(2x+1x)+(-2y+2y)=22+59}}} Group like terms.



{{{3x+0y=81}}} Combine like terms.



{{{3x=81}}} Simplify.



{{{x=(81)/(3)}}} Divide both sides by {{{3}}} to isolate {{{x}}}.



{{{x=27}}} Reduce.



------------------------------------------------------------------



{{{2x-2y=22}}} Now go back to the first equation.



{{{2(27)-2y=22}}} Plug in {{{x=27}}}.



{{{-2y=22-54}}} Subtract {{{54}}} from both sides.



{{{-2y=-32}}} Combine like terms on the right side.



{{{y=(-32)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{y}}}.



{{{y=16}}} Reduce.



So the solutions are {{{x=27}}} and {{{y=16}}}.



Which form the ordered pair *[Tex \LARGE \left(27,16\right)].



This means that the system is consistent and independent.



So the two numbers are 27 and 16.