A logistics company operates two models of cargo trucks, Model X and Model Y, to transport… — GMAT

A logistics company operates two models of cargo trucks, Model X and Model Y, to transport freight between two distribution hubs. Model X consumes $10$ liters of fuel per $100$ kilometers driven in city traffic and $6$ liters of fuel per $100$ kilometers driven on open highway. Model Y consumes $14$ liters of fuel per $100$ kilometers in city traffic and $8$ liters per $100$ kilometers on open highway. On a trip between the two hubs along a route composed entirely of city traffic segments and open highway segments, Model X consumed a total of $42$ liters of fuel, and Model Y consumed a total of $58$ liters of fuel. What is the total length, in kilometers, of the open highway segments along this route?

Cevap: 200 km

Cevap

The total length of the open highway segments along the route is 200 kilometers.

By converting the per-100-kilometer fuel consumption into per-kilometer rates, we obtain two linear equations in two variables:

0.10C + 0.06H = 42

and

0.14C + 0.08H = 58

. Solving this system by elimination yields

H = 200

kilometers for the open highway distance.

Adım Adım Çözüm

Define variables for city traffic distance and open highway distance.

Let

C

equal the distance in km of city traffic segments, and

H

equal the distance in km of open highway segments.

Establishing explicit variables allows the translation of the word problem context into algebraic equations.

Convert consumption rates to liters per kilometer and formulate total fuel equations for both trucks.

Model X equation:

0.10C + 0.06H = 42

; Model Y equation:

0.14C + 0.08H = 58

Dividing the consumption per 100 km by 100 gives the unit rates per km for city and highway driving.

Simplify the system of linear equations by eliminating decimals and common factors.

First equation:

5C + 3H = 2100

; Second equation:

7C + 4H = 2900

Simplifying equations reduces computation errors during elimination.

Solve the system using elimination to isolate the highway distance variable

H

Multiplying the equations to match coefficients of

C

(

35C + 21H = 14700

and

35C + 20H = 14500

) and subtracting gives

H = 200

Eliminating

C

directly yields the target quantity

H

without needing additional substitution steps.

Anahtar Kavram

Formulating and Solving Systems of Linear Equations from Multi-Condition Word Problems

Tahmini Süre:2m 0s

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