Answer
$(-\sqrt 8, -1)$, $(-\sqrt 8, 1)$, $(\sqrt 8, -1)$, $(\sqrt 8, 1)$
Work Step by Step
$4x^2+3y^2=35$
$5x^2+2y^2=42$
$5x^2+2y^2-(4x^2+3y^2)=42-35$
$x^2-y^2=7$
$x^2-y^2+y^2=7+y^2$
$x^2=7+y^2$
$4x^2+3y^2=35$
$4(7+y^2)+3y^2=35$
$28+4y^2+3y^2=35$
$28+7y^2=35$
$28+7y^2-28=35-28$
$7y^2=7$
$7y^2/7=7/7$
$y^2=1$
$\sqrt {y^2}= \sqrt 1$
$y= ±1$
$y=1$
$4x^2+3y^2=35$
$4x^2+3*1^2=35$
$4x^2+3*1=35$
$4x^2+3=35$
$4x^2+3-3=35-3$
$4x^2=32$
$4x^2/4=32/4$
$x^2=8$
$\sqrt {x^2} =\sqrt 8$
$x = ±\sqrt 8$
Please see the graph to show that the four points are all answers to the system.
Red equation: $4x^{2}+3y^{2}=35$
Purple equation: $5x^{2}+2y^{2}=42$
