Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { l }
2 ( x + 2 ) + 9 ( y - 4 ) = 1 \
2 ( x - 1 ) = - 9 ( y + 2 )
\end{array} \right.$$
A) $( 8,4 )$
B) $( 7 , - 2 )$
C) $( - 3,5 )$
D)The equations are dependent.
E)The system is inconsistent.

Question

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { l } 
2 ( x + 2 ) + 9 ( y - 4 ) = 1 \
2 ( x - 1 ) = - 9 ( y + 2 )
\end{array} \right.$$
A) $( 8,4 )$
B) $( 7 , - 2 )$
C) $( - 3,5 )$
D)The equations are dependent.
E)The system is inconsistent.

Quizplus · Accepted Answer

First, simplify and expand both equations: 1) $2(x + 2) + 9(y - 4) = 1$ becomes $2x + 4 + 9y - 36 = 1$, which simplifies to $2x + 9y - 32 = 0$.2) $2(x - 1) = -9(y + 2)$ becomes $2x - 2 = -9y - 18$, which simplifies to $2x + 9y + 16 = 0$.Multiplying the first equation by 1 and the second equation by 1 does not eliminate any variable when added together, indicating the system might be inconsistent or dependent. Upon closer inspection, there is no scalar multiple that makes the two equations identical (which would indicate dependence) or results in a contradiction (like $0 = 1$, which would indicate inconsistency). However, rearranging the terms and comparing the equations:1) $2x + 9y = 32$2) $2x + 9y = -16$These two equations cannot be true at the same time since they assign two different constants to the same linear combination of $x$ and $y$, indicating the system is inconsistent.

Use the Addition Method to Solve the System {2(x+2)+9(y−4)=12(x−1)=−9(y+2)\left\{ \begin{array} { l } 2 ( x + 2 ) + 9 ( y - 4 ) = 1 \\2 ( x - 1 ) = - 9 ( y + 2 )\end{array} \right.{2(x+2)+9(y−4)=12(x−1)=−9(y+2)​

Use the Addition Method to Solve the System $\left\{ \begin{array} { l } 2 ( x + 2 ) + 9 ( y - 4 ) = 1 \\2 ( x - 1 ) = - 9 ( y + 2 )\end{array} \right.$