A luxury car gets 16 mpg in the city and 24 mpg on the...

A luxury car gets $16 \mathrm{mpg}$ in the city and $24 \mathrm{mpg}$ on the highway. A sport-utility vehicle (SUV) gets $14 \mathrm{mpg}$ in the city and 20 $\mathrm{mpg}$ on the highway. Suppose that each vehicle travels $c$ miles in the city and $h$ miles on the highway. The luxury car uses 12 gal of gasoline and the SUV uses 14 gal. Solve the system of equations to determine the number of city miles driven and the number of highway miles driven. Luxury car: $\frac{1}{16} c+\frac{1}{24} h=12$ SUV: $\quad \frac{1}{14} c+\frac{1}{20} h=14$

A luxury car gets $16 \mathrm{mpg}$ in the city and $24 \mathrm{mpg}$ on the highway. A sport-utility vehicle (SUV) gets $14 \mathrm{mpg}$ in the city and 20 $\mathrm{mpg}$ on the highway. Suppose that each vehicle travels $c$ miles in the city and $h$ miles on the highway. The luxury car uses 12 gal of gasoline and the SUV uses 14 gal. Solve the system of equations to determine the number of city miles driven and the number of highway miles driven.
Luxury car: $\frac{1}{16} c+\frac{1}{24} h=12$
SUV: $\quad \frac{1}{14} c+\frac{1}{20} h=14$

Key Concepts

Application of Unit Rates and Fuel Efficiency

Unit rates, such as miles per gallon in fuel efficiency problems, allow the translation of real-world scenarios into algebraic equations. Understanding and using these rates is crucial for setting up equations that relate different quantities, like distances traveled and fuel consumed, enabling the analysis and solution of practical problems involving proportional relationships.

Elimination Method

The elimination method is a systematic approach for solving systems of linear equations by removing one variable at a time. This is achieved by aligning equations so that adding or subtracting them cancels one variable, thereby reducing the system to a simpler one which can then be solved step-by-step.

Clearing Fractions

Clearing fractions is a method used to simplify equations by eliminating denominators. When equations contain fractions, multiplying both sides by the least common denominator transforms the equations into equivalent ones with integer coefficients, making further manipulation and solution simpler and less error-prone.

System of Linear Equations

A system of linear equations involves multiple linear equations that are simultaneously satisfied. In problems like this one, the equations represent relationships between unknown quantities (such as miles driven in the city and on the highway) based on given conditions (like fuel consumption and efficiency), and solving the system reveals the specific values of these unknowns.

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