Solve the system by the addition method. $$\begin{array} { l }
2 x ^ { 2 } + 2 y ^ { 2 } = 18 \
2 y - x = 12
\end{array}$$
A) $\{ ( 9,6 ) , ( - 9 , - 6 ) \}$
B) $\{ ( 0,0 ) \}$
C) $\{ ( 18,6 ) , ( - 18,6 ) \}$
D) { }

Question

Solve the system by the addition method. $$\begin{array} { l } 
2 x ^ { 2 } + 2 y ^ { 2 } = 18 \
2 y - x = 12
\end{array}$$
A) $\{ ( 9,6 ) , ( - 9 , - 6 ) \}$
B) $\{ ( 0,0 ) \}$
C) $\{ ( 18,6 ) , ( - 18,6 ) \}$
D) { }

Quizplus · Accepted Answer

The given system of equations is:$$2x^2 + 2y^2 = 18$$$$2y - x = 12$$To solve this system by the addition method, we first solve the second equation for one of the variables, say $x$:$$2y - x = 12 \Rightarrow x = 2y - 12$$Substituting $x = 2y - 12$ into the first equation:$$2(2y - 12)^2 + 2y^2 = 18$$$$2(4y^2 - 48y + 144) + 2y^2 = 18$$$$8y^2 - 96y + 288 + 2y^2 = 18$$$$10y^2 - 96y + 270 = 0$$This quadratic equation in terms of $y$ does not match any of the given solution sets directly. To determine if there are real solutions, we can calculate the discriminant ($D = b^2 - 4ac$) of the quadratic equation:$$D = (-96)^2 - 4(10)(270) = 9216 - 10800 = -1584$$Since the discriminant is negative ($D < 0$), there are no real solutions to the system of equations. Therefore, the correct answer is D, indicating that the system has no solution.

Solve the System by the Addition Method 2x2+2y2=182y−x=12\begin{array} { l } 2 x ^ { 2 } + 2 y ^ { 2 } = 18 \\2 y - x = 12\end{array}2x2+2y2=182y−x=12​

Solve the System by the Addition Method $\begin{array} { l } 2 x ^ { 2 } + 2 y ^ { 2 } = 18 \\2 y - x = 12\end{array}$