Graph the equations to determine whether the system has any solutions. Find any solutions that exist. $$\left\{ \begin{array} { r }
2 x + y = 12 \
x ^ { 2 } + y ^ { 2 } = 36
\end{array} \right.$$
A) $( 6,0 ) \left( - \frac { 18 } { 5 } , - \frac { 24 } { 5 } \right)$
B) $( - 6,0 ) \left( \frac { 18 } { 5 } , \frac { 24 } { 5 } \right)$
C) $( 6,0 ) \left( \frac { 18 } { 5 } , \frac { 24 } { 5 } \right)$
D) $( - 6,0 ) \left( - \frac { 18 } { 5 } , - \frac { 24 } { 5 } \right)$
E)no solution exists

Question

Graph the equations to determine whether the system has any solutions. Find any solutions that exist. $$\left\{ \begin{array} { r } 
2 x + y = 12 \
x ^ { 2 } + y ^ { 2 } = 36
\end{array} \right.$$
A) $( 6,0 ) \left( - \frac { 18 } { 5 } , - \frac { 24 } { 5 } \right)$
B) $( - 6,0 ) \left( \frac { 18 } { 5 } , \frac { 24 } { 5 } \right)$
C) $( 6,0 ) \left( \frac { 18 } { 5 } , \frac { 24 } { 5 } \right)$
D) $( - 6,0 ) \left( - \frac { 18 } { 5 } , - \frac { 24 } { 5 } \right)$
E)no solution exists

Quizplus · Accepted Answer

We can solve for $y$ in the first equation to get $y=12-2x$ and substitute into the second equation: $$x^2+(12-2x)^2=36 \implies 5x^2-48x+72=0$$ Solving for $x$ using the quadratic formula we get: $$x=\frac{48 \pm \sqrt{48^2-4(5)(72)}}{10}= \frac{24 \pm 6\sqrt{21}}{5}$$ Substituting $x$ back into the first equation we get the corresponding solutions for $y$: $$x=\frac{24 + 6\sqrt{21}}{5} \implies y=\frac{24 - 2(24 + 6\sqrt{21})}{5}=-\frac{24}{5} -\frac{12\sqrt{21}}{5}$$ $$x=\frac{24 - 6\sqrt{21}}{5} \implies y=\frac{24 - 2(24 - 6\sqrt{21})}{5}=-\frac{24}{5} +\frac{12\sqrt{21}}{5}$$ Therefore, the system has two solutions: $\left(\frac{24 + 6\sqrt{21}}{5},-\frac{24}{5} -\frac{12\sqrt{21}}{5}ight)$ and $\left(\frac{24 - 6\sqrt{21}}{5},-\frac{24}{5} +\frac{12\sqrt{21}}{5}ight)$. The correct choice is $\boxed{	extbf{(C)}}$.

Graph the Equations to Determine Whether the System Has Any {2x+y=12x2+y2=36\left\{ \begin{array} { r } 2 x + y = 12 \\x ^ { 2 } + y ^ { 2 } = 36\end{array} \right.{2x+y=12x2+y2=36​

Graph the Equations to Determine Whether the System Has Any $\left\{ \begin{array} { r } 2 x + y = 12 \\x ^ { 2 } + y ^ { 2 } = 36\end{array} \right.$