Answer:
In order to have a consistent linear system represented by the augmented matrix:
[tex]\left[\begin{array}{ccc}2&-3&h\\-6&9&5\end{array}\right][/tex]
the value of h must be:
[tex]h=-\frac{5}{3}[/tex]
Step-by-step explanation:
A system is consistent if it has a solution, this solution can be unique or a set of infinite solutions.
First, you take the augmented matrix and find the equivalent row echelon form using Gaussian-Jordan elimination:
To do this, you have to multiply the 1st row by 3 and add it to the 2nd row, the resulting matrix is:
[tex]\left[\begin{array}{ccc}2&-3&h\\0&0&5+3h\end{array}\right][/tex]
Now, write the system of equations:
[tex]2x_1-3x_2=h\\0x_1+0x_2=5+3h[/tex]
The only way this system has a solution is if 5+3h=0, then, to satisfy this, the value of h must be:
[tex]h=-\frac{5}{3}[/tex]
