Final answer:
The appropriate system of linear equations is not provided by the given reference but would require a set of three equations (like those listed in the question) to solve for the variables x, y, and z.
Step-by-step explanation:
The system of linear equations that can be solved using the given information would require equations that when merged provide a solution for the variables x, y, and z. The provided equations are 1) 2x - 3y - 4z = 5, 2) 3x + 2y - 6z = 7, 3) 4x - 6y + 8z = 9, and 4) 5x + 4y - 2z = 3. These equations can potentially form a solvable system if they are linearly independent and the determinant of the matrix formed by their coefficients is non-zero.
However, when looking at the information provided in the reference, which includes equations such as 7 y = 6x + 8, 4y = 8, and y + 7 = 3x, we can see that these are independent linear equations with two variables, x and y. The question seems to be a mismatch for the reference, as the reference provides no context for solving a three-variable system of equations. The applicable approach to solve the system would be to use either substitution, elimination (linear combination), or matrix methods such as Gaussian elimination or Cramer's rule.