A municipal water treatment facility operates three types of filtration units: Model X, Mo… — GMAT

A municipal water treatment facility operates three types of filtration units: Model X, Model Y, and Model Z.

\bullet \text{ 2 Model X units, 3 Model Y units, and 1 Model Z unit together process 134,000 gallons per hour.}

\bullet \text{ 1 Model X unit, 4 Model Y units, and 2 Model Z units together process 156,000 gallons per hour.}

\bullet \text{ 3 Model X units, 1 Model Y unit, and 4 Model Z units together process 196,000 gallons per hour.}

What is the processing capacity, in thousands of gallons per hour, of a single Model Y filtration unit?

Answer: 20 thousand gallons per hour

The processing capacity of a single Model Y filtration unit is 20 thousand gallons per hour.

By setting up the system of three linear equations

2x + 3y + z = 134

x + 4y + 2z = 156

, and

3x + y + 4z = 196

, and eliminating variables systematically through substitution and combination, we find

y = 20

. Thus, a single Model Y filtration unit processes 20 thousand gallons per hour.

Set up a system of linear equations in three variables.

Let

x

y

, and

z

represent the hourly capacities (in thousands of gallons) of Model X, Model Y, and Model Z respectively.

\begin{aligned} (1) \quad 2x + 3y + z &= 134 \\ (2) \quad x + 4y + 2z &= 156 \\ (3) \quad 3x + y + 4z &= 196 \end{aligned}

Expressing the given conditions algebraically translates the word problem into a solvable linear system.

Express

x

in terms of

y

and

z

using Equation (2).

x = 156 - 4y - 2z

Equation (2) has a coefficient of 1 for

x

, making it ideal for algebraic substitution.

Substitute

x

into Equations (1) and (3) to eliminate

x

Substituting into Equation (1):

2(156 - 4y - 2z) + 3y + z = 134 \implies 312 - 8y - 4z + 3y + z = 134

312 - 5y - 3z = 134 \implies (4) \quad 5y + 3z = 178

Substituting into Equation (3):

3(156 - 4y - 2z) + y + 4z = 196 \implies 468 - 12y - 6z + y + 4z = 196

468 - 11y - 2z = 196 \implies (5) \quad 11y + 2z = 272

Reducing a 3-variable system to a 2-variable system simplifies the calculation.

Eliminate variable

z

from Equations (4) and (5) to solve for

y

Multiply Equation (4) by 2:

10y + 6z = 356

Multiply Equation (5) by 3:

33y + 6z = 816

Subtract the first result from the second:

(33y + 6z) - (10y + 6z) = 816 - 356

23y = 460 \implies y = 20

Eliminating

z

directly yields the required value of

y

, which is the capacity of Model Y.

Solving systems of three linear equations in three variables via substitution and elimination.

Question