Question

...JI Use the Gauss-Jordan method to solve the following system of equations. 3x + 7y - 2z = 0 7x - y + 3z = 1 10x + 6y + z = 1 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution is ( C I D), in the order x, y, z. (Simplify your answers.) OB. There is an infinite number of solutions. The solution is O C. There is no solution. z), where z is any real number.

Answer 1

The system of equations
`3x+7y-2z=0\\7x-y+3z=1\\10x+6y+z=1` has infinitely many solutions
`x=-(19)/(52)z+(7)/(52)\\y=(23)/(52)z=-(3)/(52)\\z= arbitrary`

Explanation:

We have the following system of equations:

`3x+7y-2z=0\\7x-y+3z=1\\10x+6y+z=1`

The augmented matrix of the system is:

`\left[\begin{array}c3&7&-2&0\\7&-1&3&1\\10&6&1&1\end{array}\right]`

The first step is to transform the augmented matrix to the reduced row echelon form as follows:

• Row Operation 1: multiply the 1st row by 1/3

`\left[\begin{array}c1&7/3&-2/3&0\\7&-1&3&1\\10&6&1&1\end{array}\right]`

• Row Operation 2: add -7 times the 1st row to the 2nd row

`\left[\begin{array}c1&7/3&-2/3&0\\0&-52/3&23/3&1\\10&6&1&1\end{array}\right]`

• Row Operation 3: add -10 times the 1st row to the 3rd row

`\left[\begin{array}c1&7/3&-2/3&0\\0&-52/3&23/3&1\\0&-52/3&23/3&1\end{array}\right]`

• Row Operation 4: multiply the 2nd row by -3/52

`\left[\begin{array}ccc1&7/3&-2/3&0\\0&1&-23/52&-3/52\\0&-52/3&23/3&1\end{array}\right]`

• Row Operation 5: add 52/3 times the 2nd row to the 3rd row

`\left[\begin{array}ccc1&7/3&-2/3&0\\0&1&-23/52&-3/52\\0&0&0&0\end{array}\right]`

• Row Operation 6: add -7/3 times the 2nd row to the 1st row

`\left[\begin{array}ccc1&0&19/52&7/52\\0&1&-23/52&-3/52\\0&0&0&0\end{array}\right]`

The second step is interpret the reduced row echelon form from this we have the following system:

`x+(19)/(52)z=(7)/(52)\\y-(23)/(52)z=-(3)/(52)\\0=0`

We can see that the last row of the system is 0 = 0 this means that the system has infinitely many solutions.

`x=-(19)/(52)z+(7)/(52)\\y=(23)/(52)z=-(3)/(52)\\z= arbitrary`