Linear relations

Definition and terminology

  • When represented graphically, the points of a linear relation can be connected by a straight line (hence the name).
  • In a linear relation, an increase by a fixed amount on the \(x\)-axis (horizontal axis) is always associated with an increase by another fixed amount on the \(y\)-axis (vertical axis).
  • If the line passes through the origin of coordinates, we say that the relation represents direct variation; otherwise, we say that it represents partial variation.
  • The point where the line meets the \(x\)-axis is called the \(\boldsymbol{x}\)-intercept. At the \(x\)-intercept, the \(y\)-coordinate is zero.
  • The point where the line meets the \(y\)-axis is called the \(\boldsymbol{y}\)-intercept. At the \(y\)-intercept, the \(x\)-coordinate is zero.

Slope

The slope of a line is a very important characteristic. It is calculated as the ratio of the change \(\Delta y\) on the \(y\)-axis to the change \(\Delta x\) on the \(x\)-axis. If we have two points \((x_1,y_1)\) and \((x_2,y_2)\), the change on the \(y\)-axis is calculated as \[\Delta y=y_2-y_1\] and that on the \(x\)-axis is calculated as \[\Delta x=x_2-x_1\] The slope is therefore calculated as \[\text{slope}=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}\]

The change in the positive direction (upwards) on the \(y\)-axis is called the rise, and the change in the positive direction (rightwards) on the \(x\)-axis is called the run. A change in the negative direction is considered a negative value. The slope of a line is therefore commonly calculated as \[\text{slope}=\dfrac{\text{rise}}{\text{run}}\]

Drag the sliders to change the slope \(m\) and the \(y\)-intercept. Notice how the run is always taken to the right to avoid confusion. In this case, the rise is positive if we go up, and negative if we go down.

The slope represents any and all of the following

  • How steep the line is; the larger the absolute value of the slope, the more steep the line. A vertical line has an undefined slope, and a horizontal line has a slope of zero.
  • The rate of change of \(y\) in relation to \(x\); i.e., how quickly \(y\) changes when \(x\) changes.
  • Whether the relation is increasing or decreasing. An increasing relation is one where the \(y\)-value increases when the \(x\)-value increases, and a decreasing relation is one where the \(y\)-value decreases when the \(x\)-value increases. An increasing linear relation has a positive slope, and a decreasing one has a negative slope.