Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l }
2 x ^ { 2 } - 3 y ^ { 2 } = 38 \
3 x ^ { 2 } + 5 y ^ { 2 } = 95
\end{array}$$
A) $\{ ( 5,2 ) , ( 2,5 ) , ( - 5 , - 2 ) , ( - 2 , - 5 ) \}$
B) $\{ ( 5 , - 2 ) , ( 5,2 ) \}$
C) $\{ ( - 5 , - 2 ) , ( - 2 , - 5 ) \}$
D) $\{ ( 5,2 ) , ( - 5,2 ) , ( 5 , - 2 ) , ( - 5 , - 2 ) \}$

Question

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l } 
2 x ^ { 2 } - 3 y ^ { 2 } = 38 \
3 x ^ { 2 } + 5 y ^ { 2 } = 95
\end{array}$$
A) $\{ ( 5,2 ) , ( 2,5 ) , ( - 5 , - 2 ) , ( - 2 , - 5 ) \}$
B) $\{ ( 5 , - 2 ) , ( 5,2 ) \}$
C) $\{ ( - 5 , - 2 ) , ( - 2 , - 5 ) \}$
D) $\{ ( 5,2 ) , ( - 5,2 ) , ( 5 , - 2 ) , ( - 5 , - 2 ) \}$

Quizplus · Accepted Answer

To solve the system of equations, we can add them together to eliminate $y^2$, or we can subtract one from the other to eliminate $x^2$. Let's add them together for simplicity:$$2x^2 - 3y^2 + 3x^2 + 5y^2 = 38 + 95$$$$5x^2 + 2y^2 = 133$$This doesn't directly lead to a simple elimination as intended, so let's correct the approach by directly solving the system using substitution or another method that directly addresses the equations given:Given:$$2x^2 - 3y^2 = 38$$ (1)$$3x^2 + 5y^2 = 95$$ (2)We can multiply the first equation by 5 and the second by 3 to align the $y^2$ terms:$$10x^2 - 15y^2 = 190$$ (3)$$9x^2 + 15y^2 = 285$$ (4)Adding equations (3) and (4) to eliminate $y^2$:$$19x^2 = 475$$$$x^2 = 25$$$$x = \pm 5$$Substituting $x = 5$ into one of the original equations to find $y$:$$2(5)^2 - 3y^2 = 38$$$$50 - 3y^2 = 38$$$$-3y^2 = -12$$$$y^2 = 4$$$$y = \pm 2$$Thus, for $x = 5$, $y$ can be $2$ or $-2$, and similarly, for $x = -5$, $y$ can also be $2$ or $-2$, leading to the solutions $(5,2)$, $(5,-2)$, $(-5,2)$, and $(-5,-2)$, which matches option D.

Give All Solutions of the Nonlinear System of Equations, Including 2x2−3y2=383x2+5y2=95\begin{array} { l } 2 x ^ { 2 } - 3 y ^ { 2 } = 38 \\3 x ^ { 2 } + 5 y ^ { 2 } = 95\end{array}2x2−3y2=383x2+5y2=95​

Give All Solutions of the Nonlinear System of Equations, Including $\begin{array} { l } 2 x ^ { 2 } - 3 y ^ { 2 } = 38 \\3 x ^ { 2 } + 5 y ^ { 2 } = 95\end{array}$