Modelling with linear relations

A linear relation is one that has a constant rate of change, which is the slope of the line. There are many linear relations in practice. The \(x\) variable (on the horizontal axis) is the independent variable, and the \(y\) (on the vertical axis) the dependent variable. Which variable is the dependent is determined by the meaning of the relation; there is no magic rule.

The slope of the line is \(\frac{\Delta y}{\Delta x}\), which means that the unit of measurement of the slope is the unit of the variable on the vertical axis divided by the unit of the variable on the horizontal axis.

• In applications where the independent variable \(t\) is time, the relation starts at \(t=0\), which is the \(y\)-intercept of the line. Therefore, in this context, the \(y\)-intercept is often called the initial value of the relation.
• In applications that involve cost, the independent variable \(n\) is the number of items and the slope is the price per item. The \(y\)-intercept is then the cost if no items are purchased; i.e. \(n=0\). Therefore, in this context, the \(y\)-intercept is often called the fixed cost.
• In applications that involve wages or commissions, there may be a salary that does not depend on work hours or sold items, in addition to hourly (or daily) wages or a commission for every sold item. The independent variable is then the number of hours worked or the number of items sold. The slope is the wages per hour, or the commission per sold item. Therefore, the \(y\)-intercept is the fixed salary in this context.
• In applications that involve a fixed budget assigned for two items, the \(x\) and \(y\) variables are the amount of those items, and their coefficients are the respective prices. The constant in the relation is the fixed budget. This gives us a form similar to the standard form, which can then be re-arranged easily.