When performing linear regression, we are trying to find linear relationships between variables. Suppose we have a cat shelter and want to know how many extra cans of cat food we need to buy after receiving new cats. A simple approach would be to find the average number of cans a cat eats per day (z) and multiply it by the number of new cats (x). This is a linear relationship: if the number of cats increases by x, make sure to buy x times z cans of cat food. Of course, other variables might affect how much a new cat eats, such as age, breed, and weight at birth. We could possibly make a better linear model by adding these as predictors.
Imagine we had measured the amount of food eaten by 85 cats, along with their weight at birth (in grams). For budgeting reasons, we wish to predict the amount of food (number of cans) a newborn cat will eat per day when it grows up. We can plot these variables against each other, as shown in Figure 5.2: