If \(x = 300 - y - z,\) what is the value of \(x?\)

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If \(x = 300 - y - z,\) what is the value of \(x?\)

(1) \(y = \dfrac{x + z}2\)

(2) \(x = \dfrac{y + z}2\)

Answer: B

Source: Manhattan GMAT

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Gmat_mission wrote:
Tue Nov 10, 2020 8:19 am
If \(x = 300 - y - z,\) what is the value of \(x?\)

(1) \(y = \dfrac{x + z}2\)

(2) \(x = \dfrac{y + z}2\)

Answer: B

Source: Manhattan GMAT
Given: x = 300 − y − z
STRATEGY: In order to solve one of the equations below for x, I'll need to eliminate the variables y and z (and leave only x's).
So, I'm going to begin by solving the given equation for a variable other than x.

Let's solve the equation for y by first adding y to both sides of the equation to get: x + y = 300 − z
Now subtract x from both sides of the equation: y = 300 − z - x

Target question: What is the value of x?

Statement 1: y = (x + z)/2
Replace y with 300 − z - x to get: 300 − z - x = (x + z)/2
It appears that we still we're unable to eliminate the variable z. But to be extra safe, let's ensure that is the case.
First, multiply both sides of the equation by 2 to get: 600 − 2z - 2x = x + z
At this point, we can see that the variable z remains, which means there are infinitely many solutions to this linear equation.
As such, statement 1 is NOT SUFFICIENT

Statement 2: x = (y + z)/2
Replace y with 300 − z - x to get: x = (300 − z - x + z)/2
Multiply both sides of the equation by 2 to get: 2x = 300 − z - x + z
Simplify the right side: 2x = 300 - x
Add x to both sides: 3x = 300
We can see that x = 100
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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