Determine the value of h such that the matrix is the augmented matrix of a linear system with infinitely many solutions.
8 -7 | 3
24 h | 9
h = ____________
To determine the value of h such that the matrix is the augmented matrix of a linear system with infinitely many solutions, we need to consider the properties of the matrix.
A linear system has infinitely many solutions if and only if the two equations in the system are dependent, meaning that one equation can be obtained by multiplying the other equation by a scalar.
In this case, the given matrix represents the augmented matrix of a linear system with two equations.
So, we need to check if one equation can be obtained by multiplying the other equation by a scalar.
Let's set up the equations based on the given augmented matrix:
8x - 7y = 3 ------(1)
24x + hy = 9 ------(2)
To check if one equation can be obtained by multiplying the other equation by a scalar, let's multiply equation (1) by 3:
3 * (8x - 7y) = 3 * 3
24x - 21y = 9 ------(3)
By comparing equation (3) with equation (2), we see that they are identical.
Therefore, for the matrix to represent an augmented matrix of a linear system with infinitely many solutions, the value of h must be such that the equations (1) and (2) are dependent, which is the case when h = -21.
So, h = -21.