algebra precalculus - How to determine linear function from "at a price (p) of 220 the demand is 180"

In my Math book I can look-up the answers for exercises. And as I do I have no idea how I would solve the following example. Probably my mind is stuck as I don't find new ways to think about the issue.

"The supply function of a good is a linear function. At a price (p) of 220 the demand is 180 units. At a price of 160 the demand is 240 units."

1. Determine the demand function.

"Also the supply function is linear. At a price of 150 the supply is 100 units and at a price of 250 the supply is 300 units".

2. Determine the supply function.

Could someone explain to me how I would approach to solve these two questions as the book doesn't provide the explanation but only the answers? Thank you.

Answer

You know that the demand function is a linear function of price $p$, say $D(p)=\alpha\cdot p+\beta$ for suitable parameters $\alpha,\beta\in\mathbb R$. From the conditions given in your problem, you know that

$$D(220)=\boldsymbol{220\alpha+\beta=180}\qquad\text{and}\qquad D(160)=\boldsymbol{160\alpha+\beta=240}.$$

From the bold equations (system of two linear equations with two variables), one simply obtains the coeffcients $\alpha=-1$, $\beta=400$ that enables you to write down the demand function as $D(p)=400-p$.

In fact, the same can be done for the supply function.