Question

Solve the system by using elimination or elimination with multiplication. Write the solution as an ordered pair leaving no spaces, or write no solution or infinitely many solutions; Example:(2,1).

3x - 2y = 9 and -3x + 2y = -9 Solve the system by using elimination or elimination with multiplication. Write the solution as an ordered pair leaving no spaces, or write no solution or infinitely many solutions; Example:(2,1).

3x - 2y = 9 and -3x + 2y = -9 @Mathematics

Answer 1

In matrix form, we have: \[\begin{bmatrix}3 & -2 \\ -3 & 2\end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix}9 \\ -9\end{bmatrix}\]As an augmented matrix, we can write this as follows.\[\left[\begin{matrix} 3 & -2 & | &9 \\ -3 & 2 & | & -9\end{matrix}\right]\]Since the second row can be expressed as a linear combination of the first row (it is simply the negative of the first row), we can thus deduce that the system is singular and does not have one unique solution. Adding the first row to the second row gives us the following.\[\left[\begin{matrix} 3 & -2 & | &9 \\ 0 & 0 & | & 0\end{matrix}\right]\]Dividing the first row by 3 gives the following.
\[\left[\begin{matrix} 1 & -\frac{2}{3} & | &3 \\ 0 & 0 & | & 0\end{matrix}\right]\]Performing back-substitution arrives us at \[x=3+\frac{2}{3}y.\]This is a general solution.