Step by step
The question
Question: for which k does 2(3x − 4) = kx − 8 have infinitely many solutions?
We find the number k that makes it happen.
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 3x is 6x.
2 times 4 is 8.
2 Compare the x parts
The x parts: 6x on the left and kx on the right.
If the x parts were different, taking one away would still leave some x.
Then we could find x: just one x would work.
We want every x to work.
So the x parts must be the same: kx = 6x.
k times x is 6 times x when k is 6.
So k = 6.
Careful!
k is in the x part.
So we compare the x parts, not the numbers.
Now the x parts are the same.
We still have to look at the numbers.
We put k = 6 in.
3 Take away the x part
We take away 6x on both sides.
They stay equal.
6x and − 6x cancel each other.
They make zero.
The x parts are gone.
Now we have −8 = −8.
When the x parts are gone, what is left is either always true or never true.
−8 = −8 is always true.
So every x works.
That is what we want.
So k = 6.
4 Check
Let's check.
We put k = 6 back in.
Try one number, x = 1.
The left side: 2(3 × 1 − 4) = 2 × (−1) = −2.
The right side: 6 × 1 − 8 = 6 − 8 = −2.
Both sides are −2.
Any other x works the same way: both sides are 6x − 8.
So every x works.
It works!
5 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: check.
Last: type it in.