Problem Solving

[GMAT math practice question]

If ( x - y - 2 )^2 + ( x + z + 3 )^2 = 0, then what is the value of y + z?

A. 0
B. 3
C. -3
D. 5
E. -5

Max@Math Revolution wrote:[GMAT math practice question]

If ( x - y - 2 )^2 + ( x + z + 3 )^2 = 0, then what is the value of y + z?

A. 0
B. 3
C. -3
D. 5
E. -5

$${\left( {x - y - 2} \right)^2} + {\left( {x + z + 3} \right)^2} = 0\,\,\,\,\,\left( * \right)$$
$$? = y + z$$
$$\left. \matrix{
{\left( {x + z + 3} \right)^2} \ge 0 \hfill \cr
{\left( {x - y - 2} \right)^2} \ge 0\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left\{ \matrix{
\,x + z + 3 = 0 \hfill \cr
\,x - y - 2 = 0 \hfill \cr} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,\,\,z - \left( { - y} \right) + 3 - \left( { - 2} \right) = 0\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = - 5$$

The correct answer is therefore (E).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Max@Math Revolution wrote:[GMAT math practice question]

If ( x - y - 2 )^2 + ( x + z + 3 )^2 = 0, then what is the value of y + z?

A. 0
B. 3
C. -3
D. 5
E. -5

Let x=0.
Plugging x=0 into (x-y-2)² + (x+z+3)² = 0, we get:
(-y-2)² + (z+3)² = 0

Since neither (-y-2)² nor (z+3)² can be negative, the resulting equation is valid only if -y-2=0 and z+3=0, with the result that y=-2 and z=-3.
Thus, y+z = -2 + (-3) = -5

The correct answer is E.

=>
Since ( x - y - 2 )^2 + ( x + z + 3 )^2 = 0, we must have x - y - 2 = 0 and x + z + 3 = 0.
So, x = y + 2 and x + z + 3 = ( y + 2 ) + z + 3 = y + z + 5 = 0.
Thus, y + z = -5.
Therefore, the answer is E.
Answer: E