Step by step
The question
Question: for which k does 3(x + 2) = 3x + k 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 2 is 6.
Careful!
The 3 multiplies every part, the 2 too.
3 times 2 is 6, not 2.
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 6 = 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 6 = k must be always true.
6 = k is true only when k is 6.
So k = 6!
5 Check
Let's check.
We put k = 6 back in.
Try one number, x = 1.
The left side: 3(1 + 2) = 3 × 3 = 9.
The right side: 3 × 1 + 6 = 3 + 6 = 9.
Both sides are 9.
Any other x works the same way: both sides are 3x + 6.
So every x works.
It works!
6 Type it in
The answer box takes 5 characters.
6 has 1.
It fits.
We type 6.
Let's look back.
First: open the brackets.
Then: compare the x parts.
Then: take away the x part.
Then: choose k.
Then: check.
Last: type it in.