Write a system of equations for each problem, and then solve...

Write a system of equations for each problem, and then solve the system. The two longest-running shows in Broadway history were The Phantom of the Opera and Cats. As of October 7 , $2008,$ there had been a total of $16,088$ performances of the two shows, with 1118 more performances of The Phantom of the Opera than Cats. How many performances were there of each show?

Write a system of equations for each problem, and then solve the system.

The two longest-running shows in Broadway history were The Phantom of the Opera and Cats. As of October 7 , $2008,$ there had been a total of $16,088$ performances of the two shows, with 1118 more performances of The Phantom of the Opera than Cats. How many performances were there of each show?

Key Concepts

Word Problem Translation

Translating a word problem into mathematical form involves identifying the relationships and conditions described in the problem and expressing them as equations. This step is essential for applying algebraic methods to find solutions.

Solution Methods

Methods such as substitution and elimination are key techniques used to solve systems of linear equations. These methods manipulate the equations to isolate and determine the values of individual variables.

Linear Equations

Linear equations are equations in which the variables appear to the first power and are not multiplied together. They form the basis of many systems of equations and are typically solved to find intersection points representing the solutions.

Variables

Variables are symbols that represent unknown quantities in a mathematical problem. They allow us to generalize relationships and create equations to represent conditions described in a word problem.

System of Equations

A system of equations consists of two or more equations that share common variables. Solving the system means finding the values for these variables that satisfy all the equations simultaneously, which is crucial in many real-world problem-solving scenarios.

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