Step by step
The question
Question: for which k does 4(x − 1) = 4x + k 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 1 is 4.
Careful!
The 4 multiplies every part, the 1 too.
4 times 1 is 4, not 1.
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 −4 = k.
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 −4 = k must be always true.
−4 = k is true only when k is −4.
So k = −4!
5 Check
Let's check.
We put k = −4 back in.
Try one number, x = 1.
The left side: 4(1 − 1) = 4 × 0 = 0.
The right side: 4 × 1 − 4 = 4 − 4 = 0.
Both sides are 0.
Any other x works the same way: both sides are 4x − 4.
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.
Last: check.