Subject: Trigonometry

Lines

I Know What a Line is, But What is a Line?

There are hundreds of definitions of a line in figurative speech, but if you want to pass your math test, it will help to remember that a line is straight and never-ending. Your teacher may try to trick you on your test and ask you how long the line is, but remember you can only measure a segment between two points. Lines have infinite length. When you draw a line on paper, you should show that the line never ends by using arrows on either end pointing far, far into the distance.

The Breakdown

When you write about lines in verbiage, you should name a line by the points along that line. See below, this line would be called line AB, or \overleftrightarrow{AB} . Later on, we will talk about how these points really are the defining features of the line.

So let's move on and talk about what your teachers want you to know about lines. If you have two lines, it's important to know if they are intersecting or parallel lines in math. I think we all know what those two terms mean, but how can you prove one or the other? Well, it all starts with the equation of a line, then we'll add on from there.

Equation of A Line

How can you tell what direction a line is traveling? Or where it intersects the x and y axis? Or how steep it is? To start, every line has its own equation to fully define it and set it apart from all other lines. All you need to find the equation of any line is either:

  • The slope of the line, m, as well as one point along the line (call it x,y)
  • Or two points on a line (call it x1,y1 & x2,y2)
  • A line either parallel or perpendicular to the line and one additional point.

The main equation for a line is:

y = mx + b

where:

m = is the slope of the line
b = the y-intercept of the line
x & y = Given any x coordinate you can find the corresponding y coordinate and vice versa.

The Slippery Slope Equation You Need to Know

We remember in the cartesian coordinate system, there are two axes, x and y which represent horizontal and vertical movement on a 2D graph. You'll need at least two points to define a line, because there are an infinite number of lines that could possibly be running through one point. Let's call those two points (x1, y1) and (x2, y2). The equation to define the slope of a line is as follows:

The slope of a line is:

m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}

or, most often referred to as:

m = \frac{rise}{run}

One thing you must be sure of is that the order of subtraction between the two points in the numerator and denominator stays the same. It doesn't matter which point goes first, so long as you reference the same point first on top and bottom (numerator and denominator).

For example if you were to take the y-value of point G first in the numerator and then you accidentally used the x-value for H first in the denominator, the solution could possibly give you a negative slope from the correct answer. Let's use two points G: (-3, -5) and H: (2, 10) for an example:

Correct Calculation:

m = \frac{10-(-5)}{2-(-3)} = \frac{15}{5} = 3

Incorrect Calculation:

m = \frac{-5-(10)}{2-(-3)} = \frac{-15}{5} = -3

So this is why it's important to keep your points x-values & y-values coordinated. When your slope is wrong, this will affect your calculated y-intercept, and the problems will follow you throughout any problem, so be sure to take your time and keep your ducks in a row, especially for a test! Practice organizing the information on your page in a way that you can commit to memory and perform the calculations the same each time.

Y-Intercept

The term y-intercept is a fancy way of saying the point where the line crosses the y-axis, or, where {x = 0} in the equation of a line. We can rearrange the equation of a line to understand for any point on the line, b=y-mx. Be sure to use the same point for both x and y values when plugging and chugging. We can find the y-intercept of a line PQ with a slope m=2/5. Let's use point P (10, -5) for our example and plug these values into the equation below.

b= y-mx = -5-\left({2\over 5}\right)*10 = -5-4 = -9

So we know for \overleftrightarrow{PQ} with slope m=2/5 that the y-intercept occurs at the point (0,-9).

Example:

Find the equation for a line passing through points (7,-5) and (-5, 3).

1) Find m: m = \frac{3-(-5)}{-5-7} = -\frac {8}{12} = -\frac {2}{3}
2) Find b: b = y-m*x = -5-\left({-2\over 3}\right) * 7 = -5 - \frac{-14}{3} = -\frac{1}{3}
3) Put it all together: y = mx + b = -\frac {2}{3}x - \frac {1}{3}

Parallel Lines

Ok, let's get real deep in this line business. Now, how do you know if two lines are parallel? The slopes, m, for each line will be the same (e.g. m = 3 for both lines, or m = \frac {1}{3} or any other constant). How can we figure this out?

Points on Both Lines: First, we would need two points on \overleftrightarrow{AB} and two points on \overleftrightarrow{CD} . Let's say these points are as follows:

A: (4,1) B: (10,4) & C: (-7,4) D: (-9,3)

Slopes of both lines: Now let's plug these points into the equations we looked at before to find out the slope of the line AB:

m_{AB}=\frac{4-1}{10-4} = \frac{3}{6} = \frac{1}{2}

Now let's compare to find out the slope of the line CD:

m_{CD}=\frac{3-4}{-9-(-7)} = \frac{-1}{-2} = \frac {1}{2}

It's safe to say these lines are parallel! Because m_{AB}=m_{CD} we know their slopes are the same, and these lines will never intersect.

Intersecting Lines

So, we understand the concept of parallel lines, which means we know that when two lines do not have the same slope, they will intersect. Most often your teachers will ask you to go a step further and understand where these two lines will intersect. We can identify that location as a point in the 2D cartesian coordinate graph.

Points on Both Lines: Again we need two points on each line, and this time we use the equation of both lines and set equal to each other to find where the two lines share one point with coordinates (x_{int}, Y_{int}) .

Our two lines will be line EF and line GH with points as follows:

E: (-3,-6) F: (5,2) & G: (-1,4) H: (2,-5)

1) Find Slopes of Both Lines:

In order to write the equation of a line, y=mx+b , you must first understand the slope m. You should do this on your own scratch piece of paper, but I've discovered the two slopes m_{EF} = 1 and m_{GH} = -3 .

2) Y-Intercepts of Both Lines:

We also need to fine the y-intercept, b, where a line cross the y-axis, and x equals zero. Again, check yourself on your own scratch paper with the correct answer. The y-intercept of line EF occurs at the point (0,-3) and the y-intercept of line GH occurs at the point (0,1)

3) Plug and Chug: Set Both Line Equations Equal to One Another

Now, plug these into an equation form for the two lines:

y_{EF}= 1x+(-3) and y_{GH} = -3x+1

We will first solve for x by assuming y is equal and setting the two equations equal to one another:

1x+(-3) = -3x+1

+3x = +3x

4x-3 = 1

+3 = +3

4x = 4

Answer: x=1

4) Now, we plug x=1 into either of the two line equations we found above:

y_{EF}= 1x - 3 = 1(1) -3

y_{EF}=1-3 = - 2

Check: So, we can check for fun by plugging the same x-value into our other equation of the line GH (if you've found the intersection point the equations should equal each other when x = 1):

y_{GH}=-3(1)+1

y_{GH}=-3+1 = -2

As you can see, when you set the x coordinate of both equations to 1 (which you solved for when setting both equations to equal each other), they both give you the answer of -2. Therefore your answer is (1, -2).

Perpendicular Lines

Congratulations! You have learned about line equation, slope, y-intercept, parallel and intersecting lines.