Step by step
The question
Question: for which k does 3(x + k) = 3x + 12 have infinitely many solutions?
Infinitely many solutions means every number works for x.
1 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 times x is 3x.
3 times k is 3k.
2 Compare the x parts
The x parts: 3x on the left and 3x on the right.
3 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 3k = 12.
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 3k = 12 must be always true.
So we find the k that makes it true.
5 Remove the 3
3k means 3 times k.
We want just k.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just k is left.
12 divided by 3 is 4.
So k = 4!
6 Check
Let's check.
We put k = 4 back in.
Try one number, x = 1.
The left side: 3(1 + 4) = 3 × 5 = 15.
The right side: 3 × 1 + 12 = 3 + 12 = 15.
Both sides are 15.
Any other x works the same way: both sides are 3x + 12.
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 3.
Last: check.