An agricultural cooperative prepares custom soil treatments using two standard nutrient fo… — GMAT

An agricultural cooperative prepares custom soil treatments using two standard nutrient formulations: Formulation X and Formulation Y. Each bag of Formulation X contains $4\text{ kg}$ of Nitrogen and $2\text{ kg}$ of Phosphate. Each bag of Formulation Y contains $2\text{ kg}$ of Nitrogen and $5\text{ kg}$ of Phosphate. A commercial farm requires a total of exactly $140\text{ kg}$ of Nitrogen and $150\text{ kg}$ of Phosphate for its crops. Which of the following pairs $(X, Y)$ represents the exact number of bags of Formulation X and Formulation Y, respectively, needed to fulfill these requirements?

25 bags of Formulation X and 20 bags of Formulation YCevap
B
20 bags of Formulation X and 25 bags of Formulation Y
C
30 bags of Formulation X and 10 bags of Formulation Y
D
15 bags of Formulation X and 30 bags of Formulation Y
E
35 bags of Formulation X and 0 bags of Formulation Y

Cevap

25 bags of Formulation X and 20 bags of Formulation Y

The correct answer is 25 bags of Formulation X and 20 bags of Formulation Y because substituting

x = 25

and

y = 20

into the system of equations satisfies both constraints:

4(25) + 2(20) = 140\text{ kg}

Nitrogen and

2(25) + 5(20) = 150\text{ kg}

Phosphate.

Adım Adım Çözüm

Set up the simultaneous equations for Nitrogen and Phosphate requirements.

Let

x

be the number of bags of Formulation X and

y

be the number of bags of Formulation Y.
Nitrogen Equation:

4x + 2y = 140

Phosphate Equation:

2x + 5y = 150

Each bag contributes a specific amount of nutrient toward the target total.

Simplify the first equation and express

y

in terms of

x

2x + y = 70 \implies y = 70 - 2x

Dividing the Nitrogen equation by 2 isolates

y

conveniently for substitution.

Substitute

y = 70 - 2x

into the Phosphate equation.

2x + 5(70 - 2x) = 150 \implies 2x + 350 - 10x = 150 \implies -8x = -200 \implies x = 25

Solving the single-variable linear equation yields the required bags of Formulation X.

Calculate

y

using the simplified expression.

y = 70 - 2(25) = 20

Determines the required bags of Formulation Y.

Anahtar Kavram

Solving Systems of Simultaneous Linear Equations

Soru

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