Find all real solutions to the system of equations using the addition method.
-$\begin{array} { r }
2 x ^ { 2 } + y ^ { 2 } = 66 \
x ^ { 2 } + y ^ { 2 } = 41
\end{array}$
A)(5, 4), (-5, -4)
B)(4, 5), (-4, 5), (4, -5), (-4, -5)
C)(4, 5), (-4, -5)
D)(5, 4), (5, -4), (-5, 4), (-5, -4)

Question

Find all real solutions to the system of equations using the addition method.
-$\begin{array} { r } 
2 x ^ { 2 } + y ^ { 2 } = 66 \
x ^ { 2 } + y ^ { 2 } = 41
\end{array}$
A)(5, 4), (-5, -4)
B)(4, 5), (-4, 5), (4, -5), (-4, -5)
C)(4, 5), (-4, -5)
D)(5, 4), (5, -4), (-5, 4), (-5, -4)

Quizplus · Accepted Answer

Subtracting the second equation from the first gives $2x^2 - x^2 = 66 - 41$, which simplifies to $x^2 = 25$. Thus, $x = \pm5$. Substituting $x^2 = 25$ into the second equation gives $25 + y^2 = 41$, leading to $y^2 = 16$ and $y = \pm4$. Therefore, the solutions are $(5, 4)$, $(5, -4)$, $(-5, 4)$, and $(-5, -4)$.

Find All Real Solutions to the System of Equations Using 2x2+y2=66x2+y2=41\begin{array} { r } 2 x ^ { 2 } + y ^ { 2 } = 66 \\x ^ { 2 } + y ^ { 2 } = 41\end{array}2x2+y2=66x2+y2=41​

Find All Real Solutions to the System of Equations Using $\begin{array} { r } 2 x ^ { 2 } + y ^ { 2 } = 66 \\x ^ { 2 } + y ^ { 2 } = 41\end{array}$