How to solve this linear system that has infinitely many solutions?
 $
  6x + 4y - 3z = - 7 \\
  6x - 6y + 5z = - 9 \\
  6x - 16y + hz = k \\
  $

Hint: We are given that the system of given three equations has infinitely many solutions, so the equation containing two unknown terms must be some composite of the other two equations. Here we use basic scaling concepts and the comparison to get the values for the unknown terms “h” and “k”.

Complete step by step solution:
Take the given three equations and entitle with the equation numbers.
 $ 6x + 4y - 3z = - 7 $ ….. (I)
 $ 6x - 6y + 5z = - 9 $ ….. (II)
 $ 6x - 16y + hz = k $ ….. (III)
Multiply equation (II) with the number , it means the number should be multiplied with all the terms of the equation on both sides of the equation.
 $ 12x - 12y + 10z = - 18 $
Now subtract equation (I) from the above equation –
 $ 6x - 16y + 13z = ( - 11) $
Now, compare the above equation with (III) equation: $ 6x - 16y + hz = k $
Comparing the like terms from the two equations, where the terms having the same variable are taken as equal and coefficient are taken as equal. The term multiplied with the variable is the coefficient of the term.
 $ \Rightarrow h = 13{\text{ and k = ( - 11)}} $
This is the required solution.
So, the correct answer is “ $ h = 13{\text{ and k = ( - 11)}} $ ”.

Note: The solutions for the system of equations can be found using the elimination method or the combination of the elimination and the substitution method. Be careful about the sign convention while subtraction and addition of like terms. When there are two opposite signs, you have to do subtraction and give sign of bigger digit while when two signs are the same, you have to do addition and give sign of the bigger digit.
Know the concepts of the consistent and inconsistent system of equations. Consistent system of equations has at least one solution whereas an inconsistent system of the equation has no solution. Observe the set of the given equations and determine the system.