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Use the Gauss-Jordan method to solve the following system of equations. 6х - 5у%3D7 12x 10y 14 - Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. The solution is (Type an ordered pair. Simplify your answer) O B. There are infinitely many solutions. The solution is (Simplify your answer. Use integers or fractions for any numbers in the expression.) y.where y is any real number O C. There is no h solution

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Answer:

The system [tex]6x-5y=7\\12x-10y=14[/tex] has infinitely many solutions [tex]x=\frac{5}{6}y +\frac{7}{6}\\y=arbitrary[/tex]

Step-by-step explanation:

We have the following system of equations:

[tex]6x-5y=7\\12x-10y=14[/tex]

The augmented matrix of the system is:

[tex]\left[\begin{array}{cc|c}6&-5&7\\12&-10&14\end{array}\right][/tex]

Transform the augmented matrix to the reduced row echelon form

  • Row Operation 1: multiply the 1st row by 1/6

[tex]\left[\begin{array}{cc|c}1&-5/6&7/6\\12&-10&14\end{array}\right][/tex]

  • Row Operation 2: add -12 times the 1st row to the 2nd row

[tex]\left[\begin{array}{cc|c}1&-5/6&7/6\\0&0&0\end{array}\right][/tex]

From the reduced row echelon form of the augmented matrix we have the corresponding system of linear equations:

[tex]x-\frac{5}{6}y=\frac{7}{6}\\0=0[/tex]

The last row of the system (0 = 0) means that the system has infinitely many solutions.

[tex]x=\frac{5}{6}y +\frac{7}{6}\\y=arbitrary[/tex]

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