Step by step
The question
Question: for which k does 3(n − 2) = kn − 6 have infinitely many solutions?
Infinitely many solutions means every number works for n.
1 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 times n is 3n.
3 times 2 is 6.
2 Compare the n parts
The n parts: 3n on the left and kn on the right.
If the n parts were different, taking one away would still leave some n.
Then we could find n: just one n would work.
We want every n to work.
So the n parts must be the same: kn = 3n.
k times n is 3 times n when k is 3.
So k = 3.
Careful!
k is in the n part.
So we compare the n parts, not the numbers.
Now the n parts are the same.
We still have to look at the numbers.
We put k = 3 in.
3 Take away the n part
We take away 3n on both sides.
They stay equal.
3n and − 3n make zero.
Cross them out.
The n parts are gone.
Now we have −6 = −6.
When the n parts are gone, what is left is either always true or never true.
−6 = −6 is always true.
So every n works.
That is what we want.
So k = 3.
4 Check
Let's check.
We put k = 3 back in.
Try one number, n = 1.
The left side: 3(1 − 2) = 3 × (−1) = −3.
The right side: 3 × 1 − 6 = 3 − 6 = −3.
Both sides are −3.
Any other n works the same way: both sides are 3n − 6.
So every n works.
It works!
Let's look back.
First: open the brackets.
Then: compare the n parts.
Then: take away the n part.
Last: check.