# This question is related to Matrix. Solve the following system by operations on the augmented matrix.2y + 3z - 7t = 3 2x + z - 2t = -5 4x + y + z - 5t = -6 y + 2z - 4t = 1

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### 1 Answer

You need to remember that you may form the augumented matrix of the system using the coefficients of variables and the constant terms such that:

`[[0,2,3,-7,3],[0,2,1,-2,-5],[4,1,1,-5,-6],[0,1,2,-4,1]]`

You may transform the second element from the fourth row into a zero multiplying the second column by 2, then subtracting the third column from the second new column such that:

`[[0,4,3,-7,3],[0,4,1,-2,-5],[4,2,1,-5,-6],[0,2,2,-4,1]]` the second original column is multiplied by 2

`[[0,1,3,-7,3],[0,3,1,-2,-5],[4,1,1,-5,-6],[0,0,2,-4,1]]` column 3 is subtracted from column 2

You need to multiply the third column by 2 such that:

`[[0,1,6,-7,3],[0,3,2,-2,-5],[4,1,2,-5,-6],[0,0,4,-4,1]]`

You need to subtract the fourth column from the third new column such that:

`[[0,1,-1,-7,3],[0,3,0,-2,-5],[4,1,-3,-5,-6],[0,0,0,-4,1]]`

You need to convert back this augumented matrix to a system of equations such that:

`y - z - 7t = 3`

`3y - 2t = -5`

`4x + y - 3z - 5t = ` -6

`-4t = 1 =gt t = -1/4`

You may substitute `-1/4` for t in equation 3y - 2t = -5 such that:

`3y+ 2/4= -5 =gt 3y = -5 - 1/2`

`3y = -11/2 =gt y = -11/6`

You may substitute `-1/4` for t and`-11/6` for y in equation y - z - 7t = 3 such that:

`-11/6- z+ 7/4= 3 =gt z= -11/6 + 7/4 - 3`

`z = (-22 + 21 - 36)/12`

`z = -37/12`

`4x= -y + 3z+ 5t - 6`

`4x = 11/6 - 37/4 - 5/4 - 6`

`4x = (22 - 126 - 72)/12`

`4x = -176/12 =gt x = -44/12 =gt x = -11/3`

**Hence, evaluating solutions to system using augumented matrix yields `x = -11/3 ; y = -11/6 ; z = -37/12 ; t = -1/4` .**