5(x - 2) = 5x - k has infinitely many solutions. Find k.

Step by step

The question

Question: for which k does 5(x − 2) = 5x − k have infinitely many solutions?

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 Compare the x parts

The x parts: 5x on the left and 5x on the right.

3 Take away the x part

We take away 5x on both sides.

They stay equal.

5x and − 5x make zero.

Cross them out.

The x parts are gone.

Now we have −10 = −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 −10 = −k must be always true.

So we find the k that makes it true.

Let's swap the two sides, so k is on the left.

Both sides are the same amount, so we can swap them.

5 Remove the −1

−k means −1 times k.

We want just k.

To remove times −1, we divide by −1.

We do it on both sides, so they stay equal.

−1 divided by −1 is 1.

So just k is left.

−10 divided by −1 is 10.

So k = 10!

6 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!

Let's look back.

First: open the brackets.

Then: compare the x parts.

Then: take away the x part.

Then: choose k.

Then: remove the −1.

Last: check.