Solve the nonlinear system of equations.
-$$\left\{ \begin{aligned}
x ^ { 2 } + y ^ { 2 } & = 625 \
x & = y ^ { 2 } - 25
\end{aligned} \right.$$
A) $( 24,7 ) , ( 24 , - 7 )$
B) $( - 25,0 ) , ( 24,49 ) , ( 24 , - 49 )$
C) $( - 25,0 ) , ( 24,7 ) , ( 24 , - 7 )$
D) $\varnothing$

Question

Quizplus · Accepted Answer

Substituting $x = y^2 - 25$ into the first equation gives $y^4 - 50y^2 + y^2 + 625 = 625$, simplifying to $y^4 - 49y^2 = 0$. Factoring gives $y^2(y^2 - 49) = 0$, so $y^2 = 0$ or $y^2 = 49$, leading to $y = 0$, $y = 7$, or $y = -7$. Substituting these values back into $x = y^2 - 25$ gives the solutions $(-25, 0)$, $(24, 7)$, and $(24, -7)$.

Solve the Nonlinear System of Equations {x2+y2=625x=y2−25\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 625 \\x & = y ^ { 2 } - 25\end{aligned} \right.{x2+y2x​=625=y2−25​

Solve the Nonlinear System of Equations $\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 625 \\x & = y ^ { 2 } - 25\end{aligned} \right.$