A sedan gets 21 mpg in the city and 27 mpg on the highway. A...

A sedan gets $21 \mathrm{mpg}$ in the city and $27 \mathrm{mpg}$ on the highway. A truck gets $15 \mathrm{mpg}$ in the city and $21 \mathrm{mpg}$ on the highway, Suppose that each vehicle travels $c$ miles in the city and $h$ miles on the highway. The sedan uses 12 gal of gasoline and the truck uses 16 gal. Solve the system of equations to determine the number of city miles driven and the number of highway miles driven. Sedan: $\quad \frac{1}{21} c+\frac{1}{27} h=12$ Truck: $\quad \frac{1}{15} c+\frac{1}{21} h=16$

A sedan gets $21 \mathrm{mpg}$ in the city and $27 \mathrm{mpg}$ on the highway. A truck gets $15 \mathrm{mpg}$ in the city and $21 \mathrm{mpg}$ on the highway, Suppose that each vehicle travels $c$ miles in the city and $h$ miles on the highway. The sedan uses 12 gal of gasoline and the truck uses 16 gal. Solve the system of equations to determine the number of city miles driven and the number of highway miles driven.
Sedan: $\quad \frac{1}{21} c+\frac{1}{27} h=12$
Truck: $\quad \frac{1}{15} c+\frac{1}{21} h=16$

Key Concepts

Systems of Linear Equations

Systems of linear equations involve multiple equations with several unknowns that are solved simultaneously. In this type of problem, each equation provides a different condition that the unknowns must satisfy, and the solution is the set of all variable values that satisfies every equation at once.

Clearing Fractions

Clearing fractions is an algebraic technique used to eliminate fractional coefficients from an equation by multiplying both sides by the least common denominator. This step simplifies the equations, making further operations such as elimination or substitution easier to perform.

Elimination Method

The elimination method is a strategy for solving systems of equations where one variable is removed by adding or subtracting equations from each other after appropriate manipulation. This process reduces the system to one with fewer variables, which can be solved more straightforwardly, ultimately leading to the solution for all variables.

Translating Word Problems to Equations

Translating word problems to equations involves interpreting the text to identify relationships and quantities that can be expressed mathematically. This process requires defining variables for unknown quantities and setting up equations based on the conditions provided in the problem, forming the bridge between real-world scenarios and algebraic expressions.

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