Step by step
The question
Question: for which k does 4(x + 3) = kx + 10 have no solution?
No solution means no number works for x.
1 Open the brackets
The 4 in front of the bracket multiplies every part inside.
So we write 4 times each part.
4 times x is 4x.
4 times 3 is 12.
2 Compare the x parts
The x parts: 4x 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 no x to work.
So the x parts must be the same: kx = 4x.
k times x is 4 times x when k is 4.
So k = 4.
Careful!
With k not 4, the x parts are different.
Then just one x works.
We want none.
Now the x parts are the same.
We still have to look at the numbers.
We put k = 4 in.
3 Take away the x part
We take away 4x on both sides.
They stay equal.
4x and − 4x make zero.
Cross them out.
The x parts are gone.
Now we have 12 = 10.
When the x parts are gone, what is left is either always true or never true.
12 = 10 is never true.
So no x works.
That is what we want.
So k = 4.
4 Check
Let's check.
We put k = 4 back in.
Try one number, x = 1.
The left side: 4(1 + 3) = 4 × 4 = 16.
The right side: 4 × 1 + 10 = 4 + 10 = 14.
16 and 14 are not equal.
With the same x parts, the left side is always 2 more.
So no x works.
It works!
Let's look back.
First: open the brackets.
Then: compare the x parts.
Then: take away the x part.
Last: check.