Question

Determine all values of h and k for which the system S 1 -3x - 3y = h -4x + ky = 10 has no solution. k= ht

Answer 1

The system will have no solution when
`k = -4` and
`h \\eq 7.5`.

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:

`-3x - 3y = h`

`-4x + ky = 10`

This system has the following augmented matrix:

`\left[\begin{array}{ccc}-3&-3&h\\-4&k&10\end{array}\right]`

The first thing i am going to do is, to help the row reducing:

`L_(1) = -(L_(1))/(3)`

Now we have

`\left[\begin{array}{ccc}1&1&-(h)/(3)\\-4&k&10\end{array}\right]`

Now I want to reduce the first row, so I do:

`L_(2) = L_(2) + 4L_(1)`

So:

`\left[\begin{array}{ccc}1&1&-(h)/(3)\\0&k+4&10 - (4h)/(3)\end{array}\right]`

From the second line, we have

`(k+4)y = 10- (4h)/(3)`

The system will have no solution when there is a value dividing 0, so, there are two conditions:

`k+4 = 0` and
`10 - (4h)/(3) \\eq 0`

`k+4 = 0`

`k = -4`

...

`10 - (4h)/(3) \\eq 0`

`(4h)/(3) \\eq 10`

`4h \\eq 30`

`h \\eq (30)/(4)`

`h \\eq 7.5`

The system will have no solution when
`k = -4` and
`h \\eq 7.5`.