Question

How to find the value of `k` that makes the system of matrix `[(15,-3,6),(-10,k,9)]` os inconsistent?

`[(15,-3,6),(-10,k,9)]`

what is inconsistent?

Answer 1

Inconsistent means there is no solution to the system of linear equations.

`=>k = 2`

Explanation:

If you envision the equations that are created from the matrix, you have:

[1] `15x - 3y = 6`
[2] `-10x + ky = 9`

To achieve inconsistency (no solution) we need to make all variables cancel each other when trying to solve the system.

If we mutiply [1] by `10` and multiply [2] by `15` we get:

[3] `150x - 30y = 60`
[4] `-150x +15ky = 135`

You can see if we were to add these equations we could solve for `y`. So, to make this inconsistent we should choose a value of `k` that makes the `y` terms cancel as well. Then the system is not solvable.

In this case, `k = 2` works. We can try this to see how it works:

[3] `150x - 30y = 60`
[5] `-150x +30y = 135`

Adding [3] and [5] now yields:

`0 = 195`

which is nonsensical. Hence, there is no solution and the system is inconsistent.

=================EDIT==================
Yes, this can be done by thinking of the linear equations as lines on a graph and trying to make the lines parallel in order to avoid intersection and create a "no solution" situation.

We have the two equations:

[1] `15x - 3y = 6`
[2] `-10x + ky = 9`

We can rewrite them in slope-intercept form:

[1] `y = 5x -2`
[2] `y = 10/k x + 9/k`

In this case, we know that lines are parallel if they have the same slope.

`5 = 10/k -> color(blue)(k = 2)`

We check the `y`-intercept to make sure there is a difference, otherwise the two lines would be the same and then there are infinite solutions.

`-2 != 9/2`

so we are good.