Step by step
The question
Question: for which k does 2(x + 5) = 2x + k have infinitely many solutions?
Infinitely many solutions means every number works for x.
1 Open the brackets
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
2 times x is 2x.
2 times 5 is 10.
Careful!
The 2 multiplies every part, the 5 too.
2 times 5 is 10, not 5.
2 Compare the x parts
The x parts: 2x on the left and 2x on the right.
3 Take away the x part
We take away 2x on both sides.
They stay equal.
2x and − 2x make zero.
Cross them out.
The x parts are gone.
Now we have 10 = 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 10 = k must be always true.
10 = k is true only when k is 10.
So k = 10!
5 Check
Let's check.
We put k = 10 back in.
Try one number, x = 1.
The left side: 2(1 + 5) = 2 × 6 = 12.
The right side: 2 × 1 + 10 = 2 + 10 = 12.
Both sides are 12.
Any other x works the same way: both sides are 2x + 10.
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.