Find the y-intercept for the following straight line.
Slope/Gradient
The slope (gradient) ultimately determines the ‘steepness’ or incline of a line, the higher the slope, the steeper the incline will be. For example, a horizontal line has a slope equal to zero while a line with an angle of 45o has a slope equal to one.
The sign (positive or negative) of the slope is very important, as it determines whether the line slopes uphill or downhill. Positive slopes (like the ones above) mean that the line slopes uphill, from left to right. Meanwhile if negative, the line slopes downhill, from right to left.
The slope of a line is usually represented as the letter ‘a’, and is defined as being the change in the y coordinate (rise) divided by the corresponding change in the x coordinate (run).
(The delta symbol, ‘Δ’, is commonly used in mathematics to indicate ‘difference’ or ‘change’)
Given two points (x1, y1) and (x2, y2), the change in the x coordinate would be the same as x2 – x1. Meanwhile, in order to find the change in the y coordinate one would have to subtract y1 from y2. Therefore, by simply substituting both quantities into the above equation, we are left with the slope formula.
Knowing this formula we can calculate the slope of any straight line, which can become very helpful if you need to find the actual equation of a line.
Examples
Given two points, (2,4) and (0,3), find the slope of the following straight line.
Given two points, (2,4) and (1,2), find the equation of the following straight line.
Vertical Line
A vertical line is a line of which is parallel to the y-axis, which simply means that all points on the line will have the same x-coordinate. A vertical line is a special case as it has no slope. Or put another way, for a vertical line the slope is undefined. The equation of a vertical line will therefore be:
Where:
Notice that the equation is independent of y. Any point on the vertical line satisfies the equation.
Perpendicular Lines
Perpendicular lines are straight lines of which intersect to form a 90o angle (right angle). Take two different lines:
Then, in this case ‘a’ and ‘c’ are the slopes of the two lines. The lines ‘L’ and ‘M’ are only perpendicular if and only if the product of their slopes is -1. In this case:
Examples
Are the following two linse perpendicular?
Do the following two lines intersect to form a 90o angle?
Appendix
Proof
Slope Formula:
Slope-Intercept Form:
Notation
Different Countries teach different "notation".
US, Canada, Egypt, Mexico, and Philippines:
UK, Australia, Bahamas, Bangladesh, Belgium, Brunei, Cyprus, Germany, Ghana, India, Indonesia, Ireland, Jamaica, Kenya, Kuwait, Malaysia, Malawi, Malta, Nepal, Netherlands, New Zealand, Nigeria, Pakistan, Singapore, Solomon Islands, South Africa, Sri Lanka, Turkey, UAE, Zambia and Zimbabwe:
Albania, Brazil, Czech Republic, Denmark, Ethiopia, France, Lebanon, Holland, Kyrgyzstan and Vietnam:
Azerbaijan, China, Finland, Russia and Ukraine:
Greece:
Italy:
Japan:
Latvia:
Romania:
Sweden:
Slovenia:
The point is that it does not matter whether the ‘slope/gradient’ is defined as an ‘m’, ‘a’ or a ‘b’, as all three letters ultimately represent the same initial thing.
See ’Notation’ in the Appendix