Question 1061574
Hi, please help me solve this by using the addition method. Thank you so much!

{{{system(x^2+y^2=16,4x^2+9y^2=36) }}}
<pre><b>

Their graphs are a circle with an ellipse inside, like this
{{{drawing(100,100,-5,5,-5,5,line(-6,0,6,0),line(0,-6,0,6),circle(0,0,4),arc(0,0,6,4) )}}} 

They do not intersect so there are no real solutions.  However 
there are imaginary solutions:

Multiply the first equation through by -4

{{{system(-4x^2-4y^2=-64,4x^2+9y^2=36) }}}

Now add them term by term

{{{ 0x^2+5y^2=-28 }}}

      {{{ 5y^2=-28 }}}

      {{{ y^2=(-28)/5}}}

      {{{ y^2=-28/5

      {{{y= "" +- sqrt(-28/5)}}}

      {{{y= "" +- i*sqrt(28/5)}}}

      {{{ y= "" +- i*sqrt(28)/sqrt(5) }}}

      {{{ y= "" +- i*sqrt(4*7)/sqrt(5) }}}

      {{{ y= "" +- 2i*sqrt(7)/sqrt(5) }}} 

      {{{ y= "" +- expr(2i*sqrt(7)/sqrt(5))*expr(sqrt(5)/sqrt(5)) }}}     

      {{{ y= "" +- 2i*sqrt(35)/5 }}}

When the first unknown you find is complicated, do not substitute
to find the other unknown.  Instead, start over and eliminate the
other unknown:

{{{system(x^2+y^2=16,4x^2+9y^2=36)}}}

Multiply the first equation through by -9

{{{system(-9x^2-9y^2=-144,4x^2+9y^2=36) }}}

Now add them term by term

{{{-5x^2+0y^2=-108 }}}

      {{{-5x^2=-108 }}}

      {{{ x^2=(-108)/(-5)}}}

      {{{ x^2=108/5

      {{{x= "" +- sqrt(108/5)}}}

      {{{x= "" +- sqrt(108/5)}}}

      {{{ x= "" +- sqrt(108)/sqrt(5) }}}

      {{{ x= "" +- sqrt(36*3)/sqrt(5) }}}

      {{{ x= "" +- 6sqrt(3)/sqrt(5) }}} 

      {{{ x= "" +- expr(6*sqrt(3)/sqrt(5))*expr(sqrt(5)/sqrt(5)) }}}     

      {{{ x= "" +- 6*sqrt(15)/5 }}} 

So there are 4 solutions:

{{{matrix(1,10,

(matrix(1,3,6*sqrt(15)/5,",",2i*sqrt(35)/5)),
",",
"",
(matrix(1,3,6*sqrt(15)/5,",",-2i*sqrt(35)/5)),
",",
"",
(matrix(1,3,-6*sqrt(15)/5,",",2i*sqrt(35)/5)),
",",
"",
(matrix(1,3,-6*sqrt(15)/5,",",-2i*sqrt(35)/5))
)}}}

Edwin</pre><b>