A typical Algebra 1 question involves determining if 2 lines are parallel, perpendicular, or neither. Students need to understand how to determine how these lines will behave, and hence be able to solve it. This type of questions lead effectively into solving system of linear equations, whether it be system of 2 equations, or x equations.
In order for the students to find out if the lines are parallel, perpendicular, or neither, they need to determine the slopes of the lines in question. The slope of the line can tell you everything you need to know about 2 or more lines. The definition of the slope of a line is defined as "rise over run", or m = (y1 - y2)/(x1 - x2). If you are looking for the numerical value of the slope, as long as you have 2 points in question, you will be able to determine the slope of the line.
For example, line A contains the points (4, 3) and (-2, -1). The slope of line A is m = (-1 - 3)/(-2 - 4) = -4/-6 = 2/3. This is calculated by using (x1, y1) = (-2, -1) and (x2, y2) = (4, 3). It does not matter if you decides to make (x1, y1) = (4, 3) and (x2, y2) = (-2, -1). The slope in this case will still be the same as it is exactly the same line, and every line should have the same slope. In this case, the slope is m = (3 - -1)/(4 - -2) = 4/6 = 2/3.
There are 2 scenarios between the slopes of 2 lines:
- Slopes are identical
When slopes between 2 lines are identical, it means that the 2 lines are parallel and they will never intersect (or meet) even if you draw the lines from NY all the way to Asia. An example of parallel lines are:
y = 3x + 4
y = 3x - 10
Note: these equations are written in the slope intercept form of y = mx + b, where m is the slope of the line.
- Slopes are different
When the slopes of the lines are different, it means that the lines will intersect at some point. Just because 2 lines visually appear to be parallel, as long as the slopes are numerically different by 0.00000000000000001 (or even smaller), they will intersect at some point in the future.
There 2 types of intersecting lines:
- When lines intersect to form 2 perpendicular lines
The formula to determine if 2 lines intersect to form perpendicular lines is to multiply both slopes of the lines and see if it equals to -1. If line A has a slope of -3/4 and line B has a slope of 4/3, then the lines are perpendicular because -3/4 * 4/3 = -1.
- When lines intersect to form a "X"
If the 2 lines have different slopes but the slopes do not multiplied to be -1, then they will intersect, just not forming perpendicular lines. For example, lines
y = 4.1x + 7
y = 4x + 7
They will intersect, but they are not perpendicular.In order for the students to become familiar with all different variations of this type of problem, they should try to do as many examples as they can.
Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T: 201-444-8020
E: glenrock@mathnasium.com
www.mathnasium.com/glenrock
