Step by step
The question
Question: for which k does 3(x − 5) = 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 5 is 15.
Careful!
The 3 multiplies every part, the 5 too.
3 times 5 is 15, not 5.
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 −15 = 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 −15 = k must be always true.
−15 = k is true only when k is −15.
So k = −15!
5 Check
Let's check.
We put k = −15 back in.
Try one number, x = 1.
The left side: 3(1 − 5) = 3 × (−4) = −12.
The right side: 3 × 1 − 15 = 3 − 15 = −12.
Both sides are −12.
Any other x works the same way: both sides are 3x − 15.
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.