The following table shows the approximate net earnings of...

The following table shows the approximate net earnings of Nintendo (in billions of yen) and Nokia (in billions of euro) in 2000,2004 , and $2008 .{ }^{8}$ $$ \begin{array}{|r|c|c|c|} \hline & \mathbf{2 0 0 0} & \mathbf{2 0 0 4} & \mathbf{2 0 0 8} \\ \hline \text { Nintendo } & 530 & 510 & 1700 \\ \hline \text { Nokia } & 30 & 30 & 52 \\ \hline \end{array} $$ Model Nintendo's net earnings as a function of Nokia's net earnings and time, using a linear function of the form $$ f(x, t)=A x+B t+C \quad(A, B, C \text { constants }) $$ where $f$ is Nintendo's net earnings (in billions of yen), $x$ is Nokia's net earnings (in billions of euro), and $t$ is time in years since 2000 . In 2007 Nokia's net earnings were about $€ 50$ billion. What does your model estimate as Nokia's net earnings that year?

The following table shows the approximate net earnings of Nintendo (in billions of yen) and Nokia (in billions of euro) in 2000,2004 , and $2008 .{ }^{8}$
$$
\begin{array}{|r|c|c|c|}
\hline & \mathbf{2 0 0 0} & \mathbf{2 0 0 4} & \mathbf{2 0 0 8} \\
\hline \text { Nintendo } & 530 & 510 & 1700 \\
\hline \text { Nokia } & 30 & 30 & 52 \\
\hline
\end{array}
$$
Model Nintendo's net earnings as a function of Nokia's net earnings and time, using a linear function of the form
$$
f(x, t)=A x+B t+C \quad(A, B, C \text { constants })
$$
where $f$ is Nintendo's net earnings (in billions of yen), $x$ is Nokia's net earnings (in billions of euro), and $t$ is time in years since 2000 . In 2007 Nokia's net earnings were about $€ 50$ billion. What does your model estimate as Nokia's net earnings that year?

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Key Concepts

Multivariable Linear Functions

This concept involves expressing a relationship where a dependent variable is modeled as a linear combination of two or more independent variables. In the context of the problem, Nintendo's net earnings are expressed as a function of both Nokia's net earnings and time, which requires determining constant coefficients that capture the linear contributions of each independent variable.

System of Linear Equations

To determine the unknown constants in the linear function (the coefficients of the independent variables and the constant term), a system of linear equations is set up using the data provided. Each data point (corresponding to a specific year) gives rise to an equation, and solving the system allows us to identify the unique values of these constants.

Extrapolation in Data Modeling

Once the model coefficients are determined, the linear function can be used to estimate values outside the original data set. In this problem, the function is used to predict Nintendo's net earnings for the year 2007 by inputting the corresponding value for time and Nokia's net earnings, illustrating the technique of extrapolation in linear modeling.

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