The form $y=mx+c$ is the known as the slope-intercept form of the equation of a straight line. This is because we can clearly read the gradient/slope ($m$) and the $y$ coordinate of the y-intercept ($c$) from this form.
The general form of the equation of a straight line is $ax+by+c=0$. It is not as easy to tell the gradient and y-intercept from this form as it requires some transposition to be done.
Given the gradient and a point on the line
Given that $m$ is the gradient of the line and $(x_1,y_1)$ is any point (that exists) on the line, we use the formula:
$$ \begin{equation}\begin{aligned} y-y_1=m(x-x_1)\\ \end{aligned}\end{equation} $$Example
Find the equation of the straight line whose slope is $5$ and passes through the point $(3,1)$.
Given two points on the line
Given the points $(x_1,y_1)$ and $(x_2,y_2)$, we use the formula:
$$ \begin{equation}\begin{aligned} \frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}\\ \end{aligned}\end{equation} $$Notice that the new formula is simply a substitution of the gradient formula, $m=\frac{y_2-y_1}{x_2-x_1}$ into the equation, $y-y_1=m(x-x_1)$.
Example
Find the equation of the straight line passing through the points $(1,2)$ and $(3,12)$.
Research Questions
You will be able to answer the following:
- What makes a straight line different from another?
- What is the slope-intercept form of the equation of a straight line?
- What is the general form of the equation of a straight line?
- How do we find the equation of a straight line when given the line’s gradient and a point on the line?
- How do we find the equation of a straight line when given two points on the line?
- What makes a circle different from another?
- What is the standard form of the equation of a circle?
- What is the general form of the equation of a circle?
- What is the set of values of $k$ for which the line $y=x+k$ intersects, touches, or does not meet a quadratic curve?