The gross federal debt y (in trillions of dollars) in year x...

The gross federal debt $y$ (in trillions of dollars) in year $x$ is approximated by
$$
y=.79 x+3.93 \quad(x \geq 3)
$$
where $x$ is the number of years after $2000 . $Find the year in which the approximate federal debt
is:
$\$ 14.2$ billion

Key Concepts

Linear Functions

A linear function is an equation that models a straight-line relationship between two variables. In such equations, the relationship is defined by a constant rate of change (slope) and a constant term (y-intercept). Linear functions are used to predict values of one variable based on changes in another, and they form the basis for many modeling situations in mathematics and applied fields.

Solving Linear Equations

This process involves isolating the variable in a linear equation in order to find its value given a particular output. When a value of the dependent variable (often y) is specified, the corresponding independent variable (often x) can be found by algebraically manipulating the equation. This technique is fundamental for analyzing models and determining input values that produce desired outputs.

Unit Conversion

Unit conversion is the process of converting quantities from one unit of measurement to another to ensure consistency throughout a calculation. When working with models that involve different units, such as billions and trillions, it is crucial to perform accurate conversions to maintain the integrity of the solutions.

Time Shifting in Models

Time shifting refers to adjusting the reference point of a time variable in a model. This is common when the variable representing time does not start at zero in the given context, such as using years after a specific base year. Aligning the model's time variable correctly simplifies interpretation and application of the model in real-world scenarios.

Transcript

Please give Ace some feedback