A specialty coffee roaster produces two custom coffee blends, Blend Alpha and Blend Beta, … — GMAT

A specialty coffee roaster produces two custom coffee blends, Blend Alpha and Blend Beta, using Arabica and Robusta beans. Each bag of Blend Alpha requires $300\text{ grams}$ of Arabica beans and $100\text{ grams}$ of Robusta beans. Each bag of Blend Beta requires $200\text{ grams}$ of Arabica beans and $200\text{ grams}$ of Robusta beans. On a given day, the roaster used a total of $14,000\text{ grams}$ of Arabica beans and $8,000\text{ grams}$ of Robusta beans to produce bags of these two blends, using all available inventory.

In the table below, select the number of bags of Blend Alpha produced and the number of bags of Blend Beta produced that are consistent with this information.

Bags of Blend Alpha produced30
Bags of Blend Beta produced25

Cevap

Bags of Blend Alpha produced = 30; Bags of Blend Beta produced = 25

By representing the number of bags of Blend Alpha as

x

and Blend Beta as

y

, the constraints form the system

3x + 2y = 140

and

x + 2y = 80

. Subtracting the equations yields

2x = 60

, so

x = 30

. Substituting

x = 30

gives

y = 25

. Thus, Blend Alpha produced is 30 and Blend Beta produced is 25.

Adım Adım Çözüm

Define variables for the unknowns.

Let

x

be the number of bags of Blend Alpha produced, and

y

be the number of bags of Blend Beta produced.

Setting up explicit variables allows formulation of linear algebraic equations.

Set up equations based on total bean usage.

For Arabica beans:

300x + 200y = 14,000

, which simplifies to

3x + 2y = 140

.
For Robusta beans:

100x + 200y = 8,000

, which simplifies to

x + 2y = 80

Translating the physical constraints into two linear equations in two variables.

Solve the system by subtracting the second equation from the first.

(3x + 2y) - (x + 2y) = 140 - 80 \implies 2x = 60 \implies x = 30

Subtracting eliminates

y

directly because both equations have identical

2y

terms.

Substitute

x = 30

into the simplified Robusta equation to find

y

30 + 2y = 80 \implies 2y = 50 \implies y = 25

Determines the remaining unknown variable.

Anahtar Kavram

Simultaneous Linear Equations in Two Variables

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