**If xy + z = x(y + z), which of the following must be true? GMAT Explanation + Additional Examples!**

If xy + z = x(y + z), which of the following must be true?

- x = 0 and y = 0
- x = 1 and y = 1
- y = 1 and z = 0
- x = 1 or y = 0
- x = 1 or z = 0

**Define the question:**

There are two things that are important to note.

*Which of the following must be true?*The keyword here is “must.” The question is not asking which of these*can*be true – so we aren’t just plugging in answer choices and seeing if the equation works out. We need to do some work to deduce what must be true about the equation.- The answer choices have slight differences. A, B, and C are all “and” statements, while D and E are “or” statements.

So, for answers A, B, or C to be correct, we’ll need both variables to have their specified value, and for D and E, either variable having its specified value on its own will suffice.

**Algebra time:**

Let’s dig into: xy + z = x(y + z).

Our first step is to simplify by distributing out the ‘x’ on the right-side.

xy + z = xy + xz.

We now have an ‘xy’ term on either side. This works out quite nicely, as we can subtract xy from both sides, making the equation much simpler:

z = xz.

Let’s get everything to one side and factor:

0 = xz – z

0 = z(x – 1)

From here, we can see that this equation is satisfied by either z = 0 or x = 1.

**Answer: E**

**Here is all of that worked out:**

**Alternate Solution:**

Going back a couple steps, let’s think more about what subtracting ‘xy’ from both sides actually means. When we end up at the simplified equation:

z = xz

There’s no ‘y’ to be found anywhere.

Since we were able to subtract out ‘y’ from the equation, we are essentially saying that the value of y does not matter. In other words: y could be anything.

The question specified that the values of the variables must be true.

As the value of y could be anything, we can effectively rule out any answer choice that specifies a value for y.

That rules out everything but E.

**Answer: E**

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