Geometry of straight lines
Posted in Maths, Core 1A Linear equation, where the highest powers of x and y are 1, will always give a straight line graph.
Gradient of a line
The gradient of two connecting points on a line is given by the equation:
Therefore the gradient of two points, (x1, y1) and (x2, y2) is:
Gradient is usually denoted by the letter m.
Finding the intercept of a line
The intercept of a line is where it cuts the y-axis and is typically denoted by the letter c
The intercept of a line can be found in one of two ways:
By solving for y
The y intercept is given when the x co-ordinate is 0. Substituting x = 0 into the equation of the line allows you to solve for the y co-ordinate.
Worked example
Find the gradient of the line 4x + 2y = 20
Substituting x = 0 into the equation gives 2y = 20, therefore y is 10.
Rearranging into y = mx + c
An equation in the form y = mx + c, where m is the gradient and c is the intercept, such as y = 2x + 8, allows you to quickly calculate the intercept, in this case 8.
If however you are given an equation in the form ax + by + c = 0 you can rearrange it into the above form.
Worked example
Find the gradient of the line 4x + 2y - 20 = 0
Firstly add 20 to both sides to leave you with 4x + 2y = 20.
Then subtract 4x from both sides to leave you with 2y = -4x + 20.
Lastly divide both sides by 2 to give y = -4x + 10. The y intercept is therefore 10.
Finding the equation of a line
The equation can be found in one of three ways.
If you are given a point on the line and the gradient the equation of the line is:
y - y1 = m(x - x1)
For example, the equation of a line with gradient 4 through the point (2, 3) is: y - 3 = 4(x - 2), which when expanded gives y - 3 = 4x - 6 and hence y = 4x - 3.
Given two points on a line the equation can be found using:
Finally given the gradient, m and intercept, c, the equation is given by y = mx + c.
Finding the mid point of a line
The mid point between two points, (x1, y1) and (x2, y2) is:
Parallel and perpendicular lines
Parallel lines have the same gradient
Perpendicular lines (lines which cross at right angles to each other) have gradients which multiply to give -1. For example, the gradient of a line perpendicular to the line with equation y = 2x + 10 is -0.1