Step by step
The question
Question: for which k does 4(x − k) = 4x have infinitely many solutions?
Infinitely many solutions means every number works for x.
1 Open the brackets
The 4 in front of the bracket multiplies every part inside.
So we write 4 times each part.
4 times x is 4x.
4 times k is 4k.
2 Compare the x parts
The x parts: 4x on the left and 4x on the right.
3 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 −4k = 0.
4 Choose k
When the x parts are gone, what is left is either always true or never true.
We want every x to work.
So −4k = 0 must be always true.
So we find the k that makes it true.
5 Remove the −4
−4k means −4 times k.
We want just k.
To remove times −4, we divide by −4.
We do it on both sides, so they stay equal.
−4 divided by −4 is 1.
So just k is left.
0 divided by −4 is 0.
So k = 0!
6 Check
Let's check.
We put k = 0 back in.
Try one number, x = 1.
The left side: 4(1 + 0) = 4 × 1 = 4.
The right side: 4 × 1 = 4.
Both sides are 4.
Any other x works the same way: both sides are 4x + 0.
So every x works.
It works!
Let's look back.
First: open the brackets.
Then: compare the x parts.
Then: take away the x part.
Then: choose k.
Then: remove the −4.
Last: check.