consistent linear system help

Question

Determine the value of h such that the matrix is the augmented matrix of a consistent linear system.

3 -7 | h

-12 28 | 6

h = ????

Answer 1

One way of thinking about a consistent linear system is that it needs to have at least one valid solution to the associated linear equations. This means that we need answer for:

3x - 7y = h

-12x + 28y = 6

We can notice that the equations are multiples of each other. If we divide the second equation by -4, we get:

3x - 7y = 1.5

If h doesn't equal 1.5, then we'll have 2 equations that contradict each other, which would be inconsistent. So, h = 1.5 for consistency.

Another approach is to use the fact that "a linear system is consistent if and only if its coefficient matrix has the same rank as does its augmented matrix" (Wikipedia). This approach should also yield the same answer.