Question 221114
what are all ordered pairs of a numbers (x,Y) which satisfy both x(x+y)=9 and y(x+y)=16


Step 1.  {{{x(x+y)=9}}} or {{{x^2+xy=9}}}.  Solving for {{{xy=9-x^2}}}


Step 2.  {{{y(x+y)=xy+y^2=16}}}.


Step 3.  Substitute {{{xy}}} from Step 1 into Step 2


{{{9-x^2+y^2=16}}}


{{{-x^2+y^2=7}}} after subtracting 9 from both sides of the equation


Step 4.  Now in Step 1, {{{x+y=9/x}}}  and substitute into Step 2 to yield


{{{y*9/x=16}}} or {{{y=16x/9}}}


Step 5.  Substitute  {{{y=16x/9}}} into Step 3.


{{{-x^2+(16x/9)^2=7}}}


{{{-x^2+256x^2/81=7}}}


Step 6.  Multiply {{{9^2=81}}} to both sides to get rid of denominator


{{{-81x^2+256x^2=81*7=567}}}


{{{175x^2=567}}}


{{{x=1.8}}} or {{{x=-1.8}}}


{{{y=16*1.8/9=3.2}}} or {{{y=-3.2}}}


Step 7.  The ordered pairs are (1.8, 3.2) and (-1.8, -3.2).


I hope the above steps were helpful. 


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And good luck in your studies!


Respectfully,
Dr J


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