Step by step
The question
Question: for which k does 6(x + k) = 6x + 3 have infinitely many solutions?
Infinitely many solutions means every number works for x.
1 Open the brackets
The 6 in front of the bracket multiplies every part inside.
So we write 6 times each part.
6 times x is 6x.
6 times k is 6k.
2 Compare the x parts
The x parts: 6x on the left and 6x on the right.
3 Take away the x part
We take away 6x on both sides.
They stay equal.
6x and − 6x make zero.
Cross them out.
The x parts are gone.
Now we have 6k = 3.
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 6k = 3 must be always true.
So we find the k that makes it true.
5 Remove the 6
6k means 6 times k.
We want just k.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6 divided by 6 is 1.
So just k is left.
We can make 3/6 simpler.
It is the same as 1/2.
So k = 1/2!
6 Check
Let's check.
We put k = 1/2 back in.
Try one number, x = 1.
The left side: 6(1 + 1/2) = 6 × 3/2 = 9.
The right side: 6 × 1 + 3 = 6 + 3 = 9.
Both sides are 9.
Any other x works the same way: both sides are 6x + 3.
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 6.
Last: check.