Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l }
x ^ { 2 } + y ^ { 2 } = 73 \
y = x ^ { 2 } - 61
\end{array} \right.$$
A) $( 3,8 ) , ( - 8,3 )$
B) $\quad ( 5,3 ) , ( 3 , - 8 )$
C) $\quad ( 8,3 ) , ( 3 , - 8 )$
D) $\quad ( 8,3 ) , ( - 8,3 )$
E) $\quad ( - 8,3 ) , ( 8,5 )$

Question

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 73 \
y = x ^ { 2 } - 61
\end{array} \right.$$
A) $( 3,8 ) , ( - 8,3 )$
B) $\quad ( 5,3 ) , ( 3 , - 8 )$
C) $\quad ( 8,3 ) , ( 3 , - 8 )$
D) $\quad ( 8,3 ) , ( - 8,3 )$
E) $\quad ( - 8,3 ) , ( 8,5 )$

Quizplus · Accepted Answer

Substitute the second equation into the first equation and solve for x:$$x^2 + (x^2 - 61)^2 = 73$$$$x^2 + x^4 - 122x^2 + 3721 = 73$$$$x^4 - 121x^2 + 3648 = 0$$$$(x^2 - 64)(x^2 - 57) = 0$$$$x^2 = 64, 57$$$$x = \pm 8, \pm \sqrt{57}$$Substitute these values of x back into the second equation to solve for y:$$y = 8^2 - 61 = 3$$$$y = (-8)^2 - 61 = 3$$$$y = (\sqrt{57})^2 - 61 = -8$$$$y = (-\sqrt{57})^2 - 61 = -8$$Therefore, the solutions to the system of equations are:$$(x, y) = (8, 3), (-8, 3)$$

Solve the System of Equations Algebraically for Real Values of X