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# If x ≠ 0 and x^2z – 4xy + y = 0, what is the value of z ?

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If x ≠ 0 and x^2z – 4xy + y = 0, what is the value of z ? [#permalink]
Bunuel wrote:
If $$x ≠ 0$$ and $$x^2z – 4xy + y = 0$$, what is the value of z ?

(1) $$z = 32x$$

(2) $$z = 4y$$

DS20378

Are You Up For the Challenge: 700 Level Questions

$$x^2z – 4xy + y = 0$$

Each statement is not sufficient alone so lets see if the two statements combined are sufficient:

$$x^2(32x) – xz + y = 0$$
$$32x^3 – 32x^2 + 8x = 0$$
$$8x(4x^2 – 32x + 1) = 0$$
$$x(2x^2 – 1)^2 = 0$$

$$x = \frac{1}{2}$$ and $$z = 16$$. SUFFICIENT.

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Re: If x ≠ 0 and x^2z – 4xy + y = 0, what is the value of z ? [#permalink]
nick1816 wrote:
Statement 1-

$$x^2z – 4xy + y = 0$$

$$x^2*32x -4xy +y = 0$$

We can't find x or z using the above equation.

Insufficient

Statement 2-

$$x^2z – 4xy + y = 0$$

$$x^2*4y – 4xy + y = 0$$

y( 4x^2 - 4x+1) = 0

y(2x-1)^2 = 0

either y= 0 or x= 1/2

If y = 0, we can find z.
If x=1/2, we can't find z.

Insufficient

Combining both equations

1. If y=0, then x=0 (reject this case)

2. If x = 1/2; x= 16

Sufficient

Bunuel wrote:
If $$x ≠ 0$$ and $$x^2z – 4xy + y = 0$$, what is the value of z ?

(1) $$z = 32x$$

(2) $$z = 4y$$

DS20378

Why when y=0 we cannot find z?
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If x ≠ 0 and x^2z – 4xy + y = 0, what is the value of z ? [#permalink]
ubertu

He wrote "we CAN find z"

Originally posted by TestPrepUnlimited on 17 Dec 2020, 21:26.
Last edited by TestPrepUnlimited on 17 Dec 2020, 21:35, edited 1 time in total.
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If x ≠ 0 and x^2z – 4xy + y = 0, what is the value of z ? [#permalink]
Bunuel wrote:
If $$x ≠ 0$$ and $$x^2z – 4xy + y = 0$$, what is the value of z ?

(1) $$z = 32x$$

(2) $$z = 4y$$

Note we have 3 variables and only one equation. The expectation is we need at least 3 equations at least to solve for z.

Statement 1:

We can kick z out of the equation and search for x instead now. The equation is $$32x^3 - 4xy + y = 0$$ which still has 2 variables, insufficient.

Statement 2:

Again replace z and find y instead now. The equation is $$4x^2y - 4xy + y = 0$$. We are able to factor this into $$y*(4x^2 - 4x + 1) = y*(2x - 1)^2 = 0$$. We cannot tell what y is when $$2x - 1 = 0$$ so insufficient.

Combined:

Combined we have $$z =32x = 4y$$, so we have $$y = 8x$$. Let us take the easier equation from statement 2, plug in $$y = 8x$$ to get $$8x*(2x - 1)^2 = 0$$. We cannot have x = 0 as the question mentions x cannot be 0. Then we can only have $$(2x - 1)^2 = 0$$ and $$2x - 1 = 0$$. Sufficient as finding x gives us y and z.

Ans: C
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Re: If x ≠ 0 and x^2z – 4xy + y = 0, what is the value of z ? [#permalink]
Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 3 variables and 1 Equation: Let the original condition in a DS question contain 3 variables and 1 Equation. Now, we know that each condition (1) and (2) would usually give us an equation each and we need 2 equations to match the numbers of variables and equations in the original condition, therefore the most likely answer is C.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find value of 'z' .

=> $$x^2z - 4xy + y = 0$$

Second and the third step of Variable Approach: From the original condition, we have 3 variables (x, y, and z) + 1 equation (x^2z - 4xy + y = 0) .To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

Let's take both conditions together.

From condition(1), z = 32x and from the condition(2) z = 4y

=> z = 32x = 4y

=> 4y = 32x or y = 8x

=> $$x^2 * 32x - 4x(8x) + (8x) = 0$$

=> $$32x^3 - 32x^2 + 8x = 0$$

=> $$8x (4x^2 - 4x + 1) = 0$$

=> $$8x (2x - 1)^2 = 0$$

=> 'x' is not equal to zero and hence, $$(2x - 1)^2 = 0$$

=> $$(2x - 1)^2 = 0$$ or x = $$\frac{1}{2}$$.

Therefore, z = 32x = 32 * $$\frac{1}{2}$$ = 16

Since the answer is unique, both conditions together are sufficient by CMT 2.

So, C is the correct answer.

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Re: If x 0 and x^2z 4xy + y = 0, what is the value of z ? [#permalink]
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Re: If x 0 and x^2z 4xy + y = 0, what is the value of z ? [#permalink]
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