You will be expected to be able to find the equation of a straight line given a variety of scenarios. This could be as simple as finding the equation of a line given two points, or it could be finding the equation of a parallel or perpendicular line given points or other relevant information. You may also be expected to represent straight lines in different formats including , , or .
Recall that the equation of a straight line is where m is the gradient and c is the y-intercept.
Line equations given two points
Recall that to find the equation of a straight line between two points, you must first find the gradient of the line that connects the two points. This gradient is usually calculated using rise over run. One of the points can then be used to calculate the y-intercept of the line.
Parallel and Perpendicular Lines
You may be required to find the equation of a straight line that is either parallel or perpendicular to another given line. This could be directly where you asked to find an equation in no uncertain terms (see Example 1 below). Alternatively, the question may be stated indirectly, where you are asked for the area of a triangle, for example, which requires finding the equations first (see Example 2 below).
It should be noted that parallel lines have the same gradient.
If the gradient of a given line is then the gradient of the line perpendicular to it is .
Note that tangents and normals are also straight lines that have these same properties. Tangents and Normals require differentiation to identify the gradient of the curve at a specific location. See Differentiation and Tangent & Normals.
Consider the straight lines that cut the y-axis at the origin, i.e. c=0 in the straight line equation. Typically, these are equations of the form where is a specified constant or a constant to find. We tend to use k instead of m when we talk about the gradient of a line that goes through the origin. k is the constant of proportion.
When the straight line takes an equation of this form, we say that y is proportional to x or .
Click here to see some examples of proportionality in physics. Note that the constant of proportionality may also be negative. This just means that the straight line will have a negative gradient and still pass through the origin.
Modelling with Straight Lines
A mathematical model is a representation of a real-life situation using mathematical concepts. For example, given a set of paired data, we might like to fit a straight line to it. See Correlation for more on this. Note that we are assuming that the paired data, when plotted, looks like a straight line (or roughly like a straight line). The fitted line, or the line of best fit when the data doesn’t fit exactly, can then be found to predict values of other unobserved pairs.
Click here to see another explanation of modelling with straight lines.
Alternatively, click here to find Questions by Topic and scroll down to all past STRAIGHT LINES questions to practice some more.
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