Step by step
The question
Question: for which k does 3x + k = kx + 5 have no solution?
No solution means no number works for x.
1 Compare the x parts
The x parts: 3x 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 = 3x.
k times x is 3 times x when k is 3.
So k = 3.
Careful!
With k not 3, 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 = 3 in.
2 Take away the x part
We take away 3x on both sides.
They stay equal.
3x and − 3x make zero.
Cross them out.
The x parts are gone.
Now we have 3 = 5.
When the x parts are gone, what is left is either always true or never true.
3 = 5 is never true.
So no x works.
That is what we want.
So k = 3.
3 Check
Let's check.
We put k = 3 back in.
Try one number, x = 1.
The left side: 3 × 1 + 3 = 3 + 3 = 6.
The right side: 3 × 1 + 5 = 3 + 5 = 8.
6 and 8 are not equal.
With the same x parts, the left side is always 2 less.
So no x works.
It works!
Let's look back.
First: compare the x parts.
Then: take away the x part.
Last: check.