Step by step
The question
Question: for which k does 4x + 5 = kx + 5 have infinitely many solutions?
Infinitely many solutions means every number works for x.
1 Compare the x parts
The x parts: 4x 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 every x to work.
So the x parts must be the same: kx = 4x.
k times x is 4 times x when k is 4.
So k = 4.
Careful!
k is in the x part.
So we compare the x parts, not the numbers.
Now the x parts are the same.
We still have to look at the numbers.
We put k = 4 in.
2 Take away the x part
We take away 4x on both sides.
They stay equal.
4x and − 4x make zero.
Cross them out.
The x parts are gone.
Now we have 5 = 5.
When the x parts are gone, what is left is either always true or never true.
5 = 5 is always true.
So every x works.
That is what we want.
So k = 4.
3 Check
Let's check.
We put k = 4 back in.
Try one number, x = 1.
The left side: 4 × 1 + 5 = 4 + 5 = 9.
The right side: 4 × 1 + 5 = 4 + 5 = 9.
Both sides are 9.
Any other x works the same way: both sides are 4x + 5.
So every x works.
It works!
Let's look back.
First: compare the x parts.
Then: take away the x part.
Last: check.