Solve the system of equations using elimination: minus, x, plus, 2, y, equals, minus, 2−x+2y=−2 and 6, x, minus, y, equals, minus, 436x−y=−43.
To solve the system of equations using elimination, we need to eliminate one variable by adding or subtracting the two equations. First, we notice that the second equation is already in the correct form: $6x - y = -43$. Let's multiply the first equation by 3 to make the coefficients of $x$ in both equations the same. \begin{align*}
3(-2+x+2y)&=3(-2)\\
-6+3x+6y&=-6\\
3x+6y&=0
\end{align*} Now the system of equations is $\begin{cases}3x+6y=0\\6x-y=-43\end{cases}$. Let's multiply the first equation by 6 to make the coefficients of $y$ in both equations the same. \begin{align*}
6(3x+6y)&=6(0)\\
18x + 36y &= 0
\end{align*} Now the system of equations is $\begin{cases}18x+36y=0\\6x-y=-43\end{cases}$. Let's subtract the second equation from the first equation. \begin{align*}
(18x+36y)-(6x-y)&=0-(-43)\\
12x+37y &= 43
\end{align*} Now we have the system of equations $\begin{cases}12x+37y=43\\6x-y=-43\end{cases}$. We can solve this system using elimination or substitution. Let's use elimination again. Let's multiply the second equation by 12 to make the coefficients of $x$ the same in both equations. \begin{align*}
12(6x-y)&=12(-43)\\
72x-12y&=-516
\end{align*} Now the system of equations is $\begin{cases}72x-12y=-516\\12x+37y=43\end{cases}$. Let's add the equations together. \begin{align*}
(72x-12y)+(12x+37y)&=-516+43\\
84x + 25y &= -473
\end{align*} Now we have the system of equations $\begin{cases}84x+25y=-473\\12x+37y=43\end{cases}$. Now, we have a system of linear equations with the same coefficients for both $x$ and $y$. We can solve this system by elimination, substituting one equation into the other, or using a matrix. Let's use elimination again. Let's multiply the second equation by 84 to make the coefficients of $x$ the same. \begin{align*}
84(12x+37y)&=84(43)\\
1008x+3108y &= 3652
\end{align*} Now the system of equations is $\begin{cases}1008x+3108y=3652\\84x+25y=-473\end{cases}$. Let's subtract the second equation from the first equation. \begin{align*}
(1008x+3108y)-(84x+25y)&=3652-(-473)\\
924x + 3083y &= 4125
\end{align*} Now we have the equation $924x + 3083y = 4125$. We now have a single linear equation with two variables. In order to completely solve this system of equations, we need another equation.