In the system of equations,   is a constant. For what value of   does the system of linear equations have infinitely many solutions?

To determine the value of the constant  for which the system of linear equations has infinitely many solutions, we need to consider the coefficients of the variables in the equations.

Let's assume we have a system of two linear equations in two variables, x and y:

Equation 1: ax + by = c1 Equation 2: dx + ey = c2

For the system to have infinitely many solutions, the two equations must be dependent, meaning one equation can be obtained by multiplying the other equation by a constant.

To check for dependency, we can compare the ratios of the coefficients of the variables in the two equations. If the ratios are equal, the equations are dependent.

Let's compare the ratios:

(a/d) = (b/e) = (c1/c2)

If the constant  is such that the ratios of the coefficients are equal, then the system of equations will have infinitely many solutions.