Using the vertical y axis in the plane a linear relationship has the form \[ y = mx + b \ \] where m and b are constants and \( m \ne 0 \). A linear relationship is called a line.
The constants m and b have the following interpretation:
Constant | Meaning |
m | The gradient of the line : the gradient of any two points on the line |
b | The y-intercept : the point where x = 0 |
If the equation of the line is \[ y = mx + b \ \] then it is called the gradient-intercept form. There are other forms for the equation of a line.
Example A line has gradient 3/2 and intercepts thr y axis at 3. To find the equation of the line:
Substitute the values given into the gradient-intercept form of a line \[ \begin{align} y = \frac{3}{2}x + 3 \end{align} \ \]
The general equation for a line in the x-y plane is \[ ax + by + c = 0 \ \] From this, the gradient-intercept form is \[ y = -\frac{a}{b} x - \frac{c}{b} \ \] so the gradient is -a/b.
Example Going back the other way, in the diagram below, a line passess through the point (0, 3) and has gradient -2.
The gradient-intercept form of the line is \[ \begin{align} y &= -2x + 3 \\ \end{align} \ \] and the general equation of the line is \[ \begin{align} 2x+y - 3 &= 0 \\ \end{align} \ \]
To show a point lies on a line, substitute the x,y values of the point into the general equation of the line and confirm the result is 0.
Example To show the point(3/2, 0) lies on the line 2x + y - 3 = 0:
Substitute the point into the equation of the line \[ \begin{align} 2(3/2) + 0 - 3 &= 3 + 0 - 3 \\ &= 0 \end{align} \ \] so the point lies on the line.
Guided Examples
If the gradient of a line is m and it passes through a point \((x_1,y_1)\) the equation of the line is \[ y-y_1 = m(x-x_1) \ \]
Example In the diagram below, a line passess through the points (0, 3) and (3/2, 0).
To find the equation of the line, first find the gradient m by calculating the gradient of the two given points: \[ \begin{align} m &= \frac{3-0}{0-3/2} \\ &= -2 \end{align} \ \] now use the point-gradient form: \[ \begin{align} y-3 &= -2(x-0) \\ y-3 &= -2x \\ 2x+y -3 &= 0 \end{align} \ \]
The x intercept is the point where y = 0. For the general equation of the line this is \[ x = -\frac{c}{a} \ \]
The y intercept is where x = 0. For the general equation of the line this is \[ y = -\frac{c}{b} \ \]
Example To find the x and y intercepts of the line 2x + 3y - 2 = 0: \[ \begin{align} \text{When } x = 0, \quad y = 2/3 \\ \text{When } y = 0, \quad x = 1 \\ \end{align} \ \]
Horizontal lines have gradient equal to zero. Their equation is \[ \ y = b \quad \]
Vertical lines have a gradient which is not defined. Their equation is \[ \ x = a \ \] where a is the x coordinate of the point where the line cuts the x axis.