Respuesta :
Answer:
The system will have no solution when [tex]k = -4[/tex] and [tex]h \neq 7.5[/tex].
Step-by-step explanation:
We can find these values by the Gauss-Jordan Elimination method.
The Gauss-Jordan elimination method is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.
We have the following system:
[tex]-3x - 3y = h[/tex]
[tex]-4x + ky = 10[/tex]
This system has the following augmented matrix:
[tex]\left[\begin{array}{ccc}-3&-3&h\\-4&k&10\end{array}\right][/tex]
The first thing i am going to do is, to help the row reducing:
[tex]L_{1} = -\frac{L_{1}}{3}[/tex]
Now we have
[tex]\left[\begin{array}{ccc}1&1&-\frac{h}{3}\\-4&k&10\end{array}\right][/tex]
Now I want to reduce the first row, so I do:
[tex]L_{2} = L_{2} + 4L_{1}[/tex]
So:
[tex]\left[\begin{array}{ccc}1&1&-\frac{h}{3}\\0&k+4&10 - \frac{4h}{3}\end{array}\right][/tex]
From the second line, we have
[tex](k+4)y = 10- \frac{4h}{3}[/tex]
The system will have no solution when there is a value dividing 0, so, there are two conditions:
[tex]k+4 = 0[/tex] and [tex]10 - \frac{4h}{3} \neq 0[/tex]
[tex]k+4 = 0[/tex]
[tex]k = -4[/tex]
...
[tex]10 - \frac{4h}{3} \neq 0[/tex]
[tex]\frac{4h}{3} \neq 10[/tex]
[tex]4h \neq 30[/tex]
[tex]h \neq \frac{30}{4}[/tex]
[tex]h \neq 7.5[/tex]
The system will have no solution when [tex]k = -4[/tex] and [tex]h \neq 7.5[/tex].
