Question 1100168
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3x^2 - 4y^2 = 32     (1)
2x^2 + 5y^2 = 52     (2)


I will use the ELIMINATION method.


Multiply eq(1) by 2 (both sides). Multiply eq(2) by 3 (both sides). 
You will get an equivalent system


6x^2 -  8y^2 =  64    (3)
6x^2 + 15y^2 = 156    (4)


Now coefficients at x^ are the same in both equations.
It was my goal to make them equal, and I reached this goal by multiplying equations.


Next step is to subtract eq(3) from eq(4).  The terms with x^2 will cancel each other, and you get an equation for SINGLE unknown y:


15y^2 - (-8y^2) = 156 - 64,   or

23y^2 = 92  ====>  y^2 = {{{92/23}}} = 4  ====>  y = +/-{{{sqrt(4)}}} = +/-2.


Case 1.  y = 2.  ====>  from eq(2)  2x^2 = 52 - 5y^2 = 52 - 5*4 = 32  ====>  x^2 = {{{32/2}}} = 16  ====>  x = +/-{{{sqrt(16)}}} = +/-4.


Case 2.  y = -2.  ====>  from eq(2)  2x^2 = 52 - 5y^2 = 52 - 5*4 = 32  ====>  x^2 = {{{32/2}}} = 16  ====>  x = +/-{{{sqrt(16)}}} = +/-4.


<U>Answer</U>.  There are 4 solutions. They are these pairs: (4,2), (4,-2), (-4,2)  and  (-4,-2).


Geometric interpretation:  two ellipses centered at the coordinate origin, intersect in 4 points.
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To see more similar solved problems of this type, look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-the-system-of-algebraic-equations-of-degree-2.lesson>Solving systems of algebraic equations of degree 2</A> 

in this site.



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Systems of equations that are not linear</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.



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