2(x - k) = 2x - 8 has infinitely many solutions. Find k. Step by Step

Step by step

The question

Question: for which k does 2(x − k) = 2x − 8 have infinitely many solutions?

2(x − k) = 2x − 8

Infinitely many solutions means every number works for x.

infinitely many solutions = every x works

1 Open the brackets

The 2 in front of the bracket multiplies every part inside.

So we write 2 times each part.

2 × x − 2 × k = 2x − 8

2 times x is 2x.

2 times k is 2k.

2x − 2k = 2x − 8

2 Compare the x parts

The x parts: 2x on the left and 2x on the right.

3 Take away the x part

We take away 2x on both sides.

They stay equal.

2x − 2k ~~− 2x~~ = 2x − 8 ~~− 2x~~

2x and − 2x make zero.

Cross them out.

The x parts are gone.

Now we have −2k = −8.

−2k = −8

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 −2k = −8 must be always true.

always true → every x works

never true → no x works

So we find the k that makes it true.

5 Remove the −2

−2k means −2 times k.

We want just k.

To remove times −2, we divide by −2.

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

−2k−2 = −8−2

−2 divided by −2 is 1.

So just k is left.

−8 divided by −2 is 4.

−8−2 = 4

So k = 4!

k = 4

6 Check

Let's check.

We put k = 4 back in.

2(x − 4) = 2x − 8

Try one number, x = 1.

The left side: 2(1 − 4) = 2 × (−3) = −6.

x = 1: left 2(1 − 4) = 2 × (−3) = −6

The right side: 2 × 1 − 8 = 2 − 8 = −6.

x = 1: right 2 × 1 − 8 = 2 − 8 = −6

Both sides are −6.

Any other x works the same way: both sides are 2x − 8.

So every x works.

It works!

same: both sides are 2x − 8

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 −2.

Last: check.

Answer