Step by step
The question
Question: for which k does 6x − 1 = kx + 4 have no solution?
No solution means no number works for x.
1 Compare the x parts
The x parts: 6x on the left and kx on the right.
If the x parts were different, taking one away would still leave some x.
Then we could find x: just one x would work.
We want no x to work.
So the x parts must be the same: kx = 6x.
k times x is 6 times x when k is 6.
So k = 6.
Careful!
With k not 6, the x parts are different.
Then just one x works.
We want none.
Now the x parts are the same.
We still have to look at the numbers.
We put k = 6 in.
2 Take away the x part
We take away 6x on both sides.
They stay equal.
6x and − 6x make zero.
Cross them out.
The x parts are gone.
Now we have −1 = 4.
When the x parts are gone, what is left is either always true or never true.
−1 = 4 is never true.
So no x works.
That is what we want.
So k = 6.
3 Check
Let's check.
We put k = 6 back in.
Try one number, x = 1.
The left side: 6 × 1 − 1 = 6 − 1 = 5.
The right side: 6 × 1 + 4 = 6 + 4 = 10.
5 and 10 are not equal.
With the same x parts, the left side is always 5 less.
So no x works.
It works!
Let's look back.
First: compare the x parts.
Then: take away the x part.
Last: check.