The amount of restaurant sales y (in billions of dollars) in...

The amount of restaurant sales $y$ (in billions of dollars) in the United States can be estimated by the linear equation $y=13.34 x+5.36,$ where $x$ is the number of years after 1970. (Source: Based on data from the National Restaurant Association) a. Use this equation to approximate the amount of restaurant sales in 2005 . b. Use this equation to approximate the year in which the amount of restaurant sales will exceed $\$ 700$ billion. c. Use this equation to approximate the amount of restaurant sales in the current year. Do you go out to eat often? Do your friends?

The amount of restaurant sales $y$ (in billions of dollars) in the United States can be estimated by the linear equation $y=13.34 x+5.36,$ where $x$ is the number of years after 1970. (Source: Based on data from the National Restaurant Association)
a. Use this equation to approximate the amount of restaurant sales in 2005 .
b. Use this equation to approximate the year in which the amount of restaurant sales will exceed $\$ 700$ billion.
c. Use this equation to approximate the amount of restaurant sales in the current year. Do you go out to eat often? Do your friends?

Key Concepts

Extrapolation and Solving Linear Equations/Inequalities

Extrapolation refers to the process of using a linear model to predict values outside the range of the observed data, while solving linear equations or inequalities involves isolating the variable of interest. In practical scenarios, this includes determining when the dependent variable exceeds a specified threshold by setting up and solving an inequality, an important method for forecasting future outcomes based on current trends.

Function Evaluation Through Substitution

Evaluating a linear function involves substituting a given value for the independent variable to calculate the corresponding value of the dependent variable. This technique is crucial for making predictions and understanding the specific impact of changes in the independent variable on the outcome, such as determining sales for a particular year.

Linear Equation Modeling

This concept involves representing relationships between two variables with a linear equation of the form y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept. This model is used to approximate or predict outcomes based on trends observed in data over time or through other quantitative measures.

Slope and Intercept Interpretation

The slope (m) and the intercept (b) in a linear equation carry significant meaning. The slope indicates the rate of change of the dependent variable with respect to the independent variable, while the y-intercept represents the value of the dependent variable when the independent variable is zero. Understanding these components is essential for explaining how changes in one variable affect the other and for providing insights into the underlying trend.

Transcript

00:01 This is the information given to you and for part a, you want to find out how much money or like the number of sales and billions after the year 2005.

00:15 Remember, this is for after 1970.

00:18 So you need to know for 2005 how many years after this.

00:23 So you're going to just subtract in this case.

00:28 So this is 35 years after.

00:37 So instead of x, you're going to use 3 .35.

00:41 So this is 13 .34 times 35 plus 5 .36.

00:48 And remember, we're talking about billions.

00:51 So this is 1334 times 35.

00:56 This is 466.

01:00 Point nine plus 536 so then i'm just going to add those two that gives me 472 26 so in 2005 there's or there are there were 472 or 272 billion in restaurant sales.

01:54 For question b, do you want to estimate in which year there's going to, the sales are going to be more than 700 billion? so you are being told what y is.

02:10 Let me do another color.

02:12 So why you want it to be 700 billion.

02:15 So remember the y is the billion.

02:17 The x is the year.

02:19 You don't know when is that going to happen.

02:21 So you're going to still use the formula.

02:25 This is x and this is 5 .36.

02:30 So then i subtract because i want to leave the x alone...