Step by step
The question
Question: for which k does 5(x − 2) = 5x + k 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 5 in front of the bracket multiplies every part inside.
So we write 5 times each part.
5 times x is 5x.
5 times 2 is 10.
Careful!
The 5 multiplies every part, the 2 too.
5 times 2 is 10, not 2.
2 Take away the x part
The x parts are the same: 5x on each side.
We take away 5x on both sides.
They stay equal.
5x and − 5x cancel each other.
They make zero.
The x parts are gone.
Now we have −10 = k.
So after step 2: we took away 5x, and now we have −10 = k.
3 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!
4 Check
Let's check.
We put k = −10 back in.
Try one number, x = 1.
The left side: 5(1 − 2) = 5 × (−1) = −5.
The right side: 5 × 1 − 10 = 5 − 10 = −5.
Both sides are −5.
Any other x works the same way: both sides are 5x − 10.
So every x works.
It works!
5 Type it in
A negative answer gets one more character, for the minus sign.
The answer box takes 6 characters.
−10 has 3.
It fits.
We type −10.
Let's look back.
First: open the brackets.
Then: take away the x part.
Then: choose k.
Then: check.
Last: type it in.