Write a system of equations and solve. In 2005, The Aviator...

Write a system of equations and solve.
In $2005,$ The Aviator was nominated for four more Academy Awards than the movie Finding Neverland. Together they received 18 nominations. How many Academy Award nominations did each movie receive? (Source; Chicago Tribune, $1 / 26 / 05$ )

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations involving the same set of unknown variables. The goal is to find values for these variables that satisfy all the equations simultaneously, making this method a fundamental tool for solving problems involving relationships between quantities.

Variable Assignment and Representation

Assigning variables to represent unknown quantities is a crucial step in algebra. This process allows the translation of real-world problems into mathematical language, providing a clear framework for setting up equations based on the relationships described in the problem.

Translating Verbal Descriptions into Equations

This concept involves converting the narrative or descriptive parts of a problem into algebraic equations. It requires carefully identifying the relationships and differences stated, then expressing these in mathematical terms, which is essential for applying systematic problem-solving techniques.

Substitution Method

The substitution method is one technique used to solve systems of equations. It involves solving one of the equations for one variable in terms of the others and then substituting this expression into the remaining equations. This method simplifies the system, reducing it to a single equation with one unknown, which can then be solved directly.

Solution Verification

Verification involves checking the calculated solutions by substituting them back into the original equations. This step ensures that the solutions are correct and satisfy all the conditions of the problem, providing confidence in the result.

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