1.1 A Linear System of Equations

Suppose you go to the supermarket with a certain budget in mind. You love mangoes and your partner loves avocados, and you want to buy as much of these as your 20 bill budget allows. Checking the prices, you see each mango costs 1 bill and each avocado costs 1 bill. Well, this is easy; since each item is exactly 1 bill, you can get a max of 20. To be fair to your partner, you decide to get equal amounts of 10 of each fruit. If we wanted to explicitly write this mathematically, we could do something as follows:

\[(x\times mangoes) \times 1\,bill + (y \times avocados) \times 1\, bill= 20 \, bills\]

The \(x\) variable represents the number of mangoes and the \(y\) variable represents the number of avocados. Note that we have different variables because the number of each fruit does not have to be the same. Here, you might have a gut feeling that we cannot solve this equation because we only have one equation and more than one unknown. That would be true and, that is something we will discuss extensively later. The part where we said “to be fair …” is where we specify we want the number of fruits to be equal. To represent this mathematically, we set:

\[x = y\]

Now, we have what is called a system of equations:

\[(x\times mangoes) \times 1\,bill + (y \times avocados) \times 1\, bill= 20 \, bills\] \[x = y\]

We call it a system to indicate that we are considering all the equations as a whole rather than individually. That means a variable shares the same value across all equations. For example, if \(x\) equals \(1\) in one equation, then it equals \(1\) in all the other equations.

For easier reading and a bit more abstraction, we could rewrite the two equations as follows:

\[x + y = 20\] \[ x = y \]

We also call this a linear system. One reason this is called linear is because these equations represent lines graphically but, we will also discuss the meaning of linearity more abstractly in a later chapter.