7y + k = 7y - 3 has infinitely many solutions. Find k.

Step by step

The question

Question: for which k does 7y + k = 7y − 3 have infinitely many solutions?

Infinitely many solutions means every number works for y.

1 Compare the y parts

The y parts: 7y on the left and 7y on the right.

2 Take away the y part

We take away 7y on both sides.

They stay equal.

7y and − 7y make zero.

Cross them out.

The y parts are gone.

Now we have k = −3.

3 Choose k

When the y parts are gone, what is left is either always true or never true.

We want every y to work.

So k = −3 must be always true.

k = −3 is true only when k is −3.

So k = −3!

4 Check

Let's check.

We put k = −3 back in.

Try one number, y = 1.

The left side: 7 × 1 − 3 = 7 − 3 = 4.

The right side: 7 × 1 − 3 = 7 − 3 = 4.

Both sides are 4.

Any other y works the same way: both sides are 7y − 3.

So every y works.

It works!

Let's look back.

First: compare the y parts.

Then: take away the y part.

Then: choose k.

Last: check.