A chemical processing facility operates two purification units, Unit A and Unit B. Unit A … — GMAT

A chemical processing facility operates two purification units, Unit A and Unit B.

Unit A processes a liquid feed stream consisting of $x$ liters of Compound X and $y$ liters of Compound Y per hour. During an 8-hour operational cycle, Unit A processes a total of $360$ liters of feed stream.

Unit B processes Compound X at twice the hourly rate of Unit A ( $2x$ liters per hour) and Compound Y at three times the hourly rate of Unit A ( $3y$ liters per hour). During a 5-hour operational cycle, Unit B processes a total of $510$ liters of feed stream.

Based on the information provided, select for Unit A Compound X Rate the number of liters of Compound X processed by Unit A per hour, and select for Unit A Compound Y Rate the number of liters of Compound Y processed by Unit A per hour. Make exactly one selection in each column.

Unit A Compound X Rate (liters/hour)33
Unit A Compound Y Rate (liters/hour)12

Cevap

The hourly processing rate of Compound X for Unit A is 33 liters per hour, and the hourly processing rate of Compound Y for Unit A is 12 liters per hour.

Formulating equations from the operational cycles yields

8(x + y) = 360 \implies x + y = 45

and

5(2x + 3y) = 510 \implies 2x + 3y = 102

. Expressing

y

45 - x

and substituting into the second equation gives

2x + 3(45 - x) = 102

, which simplifies to

-x + 135 = 102 \implies x = 33

. Substituting

x = 33

back gives

y = 12

. Therefore, Unit A processes 33 liters of Compound X per hour and 12 liters of Compound Y per hour.

Adım Adım Çözüm

Set up the first linear equation using the total volume processed by Unit A during its 8-hour operational cycle.

8(x + y) = 360 \implies x + y = 45

Unit A processes

x + y

liters per hour, so over 8 hours it processes

8(x + y) = 360

liters.

Set up the second linear equation using the total volume processed by Unit B during its 5-hour operational cycle.

5(2x + 3y) = 510 \implies 2x + 3y = 102

Unit B processes

2x + 3y

liters per hour, so over 5 hours it processes

5(2x + 3y) = 510

liters.

Solve the system of two simultaneous linear equations for

x

and

y

x = 33

and

y = 12

From equation 1,

y = 45 - x

. Substituting into equation 2 gives

2x + 3(45 - x) = 102 \implies 2x + 135 - 3x = 102 \implies -x = -33 \implies x = 33

. Thus

y = 45 - 33 = 12

Anahtar Kavram

Solving systems of simultaneous linear equations by substitution or elimination in a Two-Part Analysis framework.

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