Criminology. City growth and the number of burglaries for a...

Criminology. City growth and the number of burglaries for a certain city are related by a linear equation. Records show that 575 burglaries were reported in a year when the local population was $77,000$ and that the rate of increase in the number of burglaries was 1 for every 100 new residents. a. Using the variables $p$ for population and $B$ for burglaries, write an equation (in slope-intercept form) that police can use to predict future burglary statistics. b. How many burglaries can be expected when the population reaches $110,000 ?$

Criminology. City growth and the number of burglaries for a certain city are related by a linear equation. Records show that 575 burglaries were reported in a year when the local population was $77,000$ and that the rate of increase in the number of burglaries was 1 for every 100 new residents.
a. Using the variables $p$ for population and $B$ for burglaries, write an equation (in slope-intercept form) that police can use to predict future burglary statistics.
b. How many burglaries can be expected when the population reaches $110,000 ?$

Key Concepts

Linear Equations

A linear equation represents a straight-line relationship between two variables, typically expressed in slope-intercept form (y = mx + b). In this form, 'm' is the slope which indicates the rate at which the dependent variable changes with respect to the independent variable, and 'b' is the y-intercept, representing the starting value when the independent variable is zero.

Slope

The slope of a linear equation quantifies how much the dependent variable increases or decreases for a unit change in the independent variable. It is a measure of the rate of change and is critical for understanding how the two variables are related, especially when making predictions based on incremental changes.

Intercept

The intercept in a linear equation provides the value of the dependent variable when the independent variable is at its zero point. It serves as the baseline or starting value from which changes are measured and is essential in formulating the complete equation for predictions.

Prediction Using Linear Models

Using a linear model allows one to predict the value of the dependent variable for any given value of the independent variable by substituting the value into the equation. This is particularly useful in practical scenarios, such as forecasting trends based on historical data.

Transcript

00:01 So this problem talks about that in a certain city, the growth of the city can be related to the number of burglaries using a linear equation.

00:10 When the population was 77 ,000, there were 575 burglaries reported, with a rate of increase of burglaries as one for every 100 new residents.

00:20 We're going to let p represent the population and b represent the number of burglarities.

00:26 We're going to write an equation in slope -intercept form that police could use to determine future burglaries to take.

00:32 So the first thing i noticed is that we're talking about slope intercept form, which is normally written in y equals mx plus b.

00:44 But we are not being asked to use the variables x and y.

00:48 We're being asked to use the variables p and b.

00:50 So before i go any further, i want to make sure that i put these variables in the right place.

00:55 Should it be p equals m capital b plus b or should it be capital b equals m p plus b? so when it comes to x and y, i know that x, backwards, i know that y depends on x.

01:16 Y is my dependent variable and x is my independent variable.

01:21 Now in this case, it's the number of burglaries that depends on the population.

01:30 So when i'm matching these things up, that means that y can be substituted, with capital b and x can be substituted with p.

01:42 So instead of y equals mx plus b, we can write this as b equals mp plus b.

01:51 So m still represents my slope.

01:54 B instead of representing my y intercept is gonna represent my b intercept.

01:59 P represents population and b represents burglaries.

02:03 So that's where i wanna start.

02:06 So my next objective would be to find my slope.

02:08 Now, normally to find slope, i do change in y over change in x.

02:16 But in this case, if i match up those variables again, that would be change in burglaries over a change in population...

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