Linear equation with a constant term

​Mikhail has a summer job as an ice cream vendor. He has paid 500 kroner ever day, and on top of that he also gets a one kroner bonus for every ice cream he sells. One kroner extra per ice cream. That sounds like a proportional salary. The more he sells, the more he earns.

Let's draw Mikhail's salary in the coordinate system. Let the y-axis represent how much Mikhail earns, and the x axis represent the number of ice creams sold. The bonus is one kroner per ice cream. The bonus y equals one times the number of ice cream's x. Since this is 1x, we do not need to write one down.

But that's just the bonus. Mikhail has a base salary as well, of 500 kroner a day, regardless of the number of ice cream sold. If it rains and Mikhail doesn't sell any ice cream, he will still earn 500 kroner that day. Here we have 500 kroner on the y-axis and zero ice creams. From here, the salary increases by one kroner for every ice cream sold.

We move the line up here. The line now describes the connection between how many ice creams Mikhail sells and how much he earns. If we write it as an equation, we get the salary, y, equals the number the ice creams, x, plus 500 from base salary. One sunny day, Mikhail sells 200 ice creams. He then receives his base salary of 500 kroner and 200 more as a bonus, 700 in total.

The base salary of 500 kroner does not change, regardless of how many ice creams he sells. It is constant. We therefore write the equation like this, y equals kx plus m. K is the coefficient that causes y to change together with x. It gives the line slope.

M is a constant. It moves the line line up or down. Maria also sells ice cream, but she does not get a bonus based on how much she sells. Instead, she gets 600 kroner a day as base salary. We can say that the equation that describes her salary only has a constant.

This is what it looks like. If Maria sells no ice cream, she earns 600. If Maria sells 200 ice creams, she still earns 600 kroner. If she sells 1,000 ice creams, she still earns 600 kroner. The salary is constant.

If we write this as an equation, it looks like this. The salary y, equals zero kroner as a bonus, times the number of ice creams, plus 600 kroner of base salary. Zero times x is zero. So, we can remove that completely. The equation then takes the form, y equals m.

This is a straight line, parallel to the x-axis, since the value of y is not affected at all by the change in x.