Lines in Coordinate Geometry

Straight lines in coordinate geometry are the same idea as in regular geometry, except that they are drawn on a coordinate plane and we can do more with them.

How to define a line

Consider the line in Fig 1. How would I define that particular line? What information could I give you over the phone so that you could draw the exact same line at your end?

Fig 1. How to define this line?

There are three ways commonly used in coordinate geometry:

Give the coordinates of any two points on the line
Give the coordinates of one point on the line, and the slope of the line
Give an equation that defines the line.

It does not matter whether we are talking about a line, ray or line segment. In all cases any of the above three methods will provide enough information to define the line exactly.

Using two points

In Fig 2, a line is defined by the two points A and B. By providing the coordinates of the two points, we can draw the line. No other line could pass through both these points and so the line they define is unique. I could call you on the phone and say "Draw a line through (9,9) and (17,4)" and you could reconstruct it perfectly on your end.

Fig 2. A,B define a unique line

For an interactive demonstration of lines defined by two points, see

Using one point and the slope

Fig 3. Point and slope define the line

The other common method is the give you the coordinates of one point and the slope of the line. For now, you can think of the slope as the direction of the line. So once you know that a line goes through a certain point, and which direction it is pointing, you have defined one unique line.

In Fig 3, we see a line passing through the point A at (14,23). We also see that its slope is +2 (which means it goes up 2 for every one across). with these two facts we can establish a unique line.

The value of the slope is usually denoted by the letter m. For more on slope and how to determine it see Slope of a Line.

Equation of a line

Once you have defined a line using the point-slope method, you can write algebra equations that describe the line. By applying algebraic processes to these equations we can solve problems that are otherwise difficult. These and many other graphing techniques are covered in the algebra volume, but the general idea is described here in Coordinate Geometry.

There are two types of equation commonly used to describe a line:

Slope-intercept (the most common). Described in Equation of a line (Slope-Intercept)
Point-slope. Described in Equation of a line (Point-Slope).

Both forms are really both variations on the same idea. In both cases you need to know the coordinates of one point, and the slope of the line.

In the slope-intercept form, the given point is always on the y-axis and you supply the y-coordinate of that point (Its x-coordinate is always zero).
In the point-slope form, you can use any point.

The place where the line crosses the y-axis is called the intercept, and is commonly denoted by the letter b. For more on this see Intercept of a line.

y = m(x-P_x) + P_y

Fig 4. Point-slope

y = mx + b

Fig 5. Slope-Intercept

If you look closely at the two formulae, you can see that they are quite similar. If you take the point-slope version in Fig 4 and choose to put P on the y-axis, its x coordinate is zero, and its y-coordinate is the same as the intercept. If you substitute those things you will get the slope-intercept formula on the right in Fig 5.

What are the equations used for?

You can use them to actually plot the line: Take various values of x, and then use the equation to find the corresponding values of y. Plot the pairs to graph the line.
If you know just one coordinate of a point on the line, you can find the other.

For more Information

The following pages expand on each of the concepts on this page: