Elimination Method:

The equations of linear system are .

Neither variable has a common coefficient.The coefficient  of the y - variables are 5 and 2 and their least common multiple is 10, so multiply each equation by the value  that will make the y - coefficient 10.

To get two equations that contain opposite terms multiply the first equation by 2 and multiply the second equation by 5.

Write the equations in column form and add the corresponding columns to eliminate y - variable.

.

The resultant equation has contain one variable x, so the linear system has one solution.

The equations of linear system are .

Neither variable has a common coefficient.The coefficient  of the y - variables are 5 and 2 and their least common multiple is 10, so multiply each equation by the value  that will make the y - coefficient 10.

To get two equations that contain opposite terms multiply the first equation by negative 2 and multiply the second equation by 5.

Write the equations in column form and add the corresponding columns to eliminate y - variable.

.

The statement is false, so the linear system has no solutions.

The equations of linear system are .

Neither variable has a common coefficient.The coefficient  of the y - variables are 5 and 2 and their least common multiple is 10, so multiply each equation by the value  that will make the y - coefficient 10.

To get two equations that contain opposite terms multiply the first equation by negative 2 and multiply the second equation by 5.

Write the equations in column form and add the corresponding columns to eliminate y - variable.

.

The statement is false, so the linear system has no solutions.

The equations of linear system are .

Neither variable has a common coefficient.The coefficient  of the y - variables are 5 and 2 and their least common multiple is 10, so multiply each equation by the value  that will make the y - coefficient 10.

To get two equations that contain opposite terms multiply the first equation by negative 2 and multiply the second equation by 5.

Write the equations in column form and add the corresponding columns to eliminate y - variable.

.

The statement is false, so the linear system has infinitely many solutions.

The option D is correct answer.