Lines Parallel to the Coordinate Axes

As we learned in a previous lesson

a line parallel to the x-axis has an undefined slope

a line parallel to the y-axis has a slope of zero.

So how do we write the equation of a line parallel to one of the coordinate axes?

The answer is simple when we remember that

the equation of a line is an algebraic statement that is true at every point on the line.

Let's look at a diagram and the answer will be obvious.

Example

(a) Write the equation of the line through A( – 2, 3) and B(–2, – 20).

(b) Write the equation of the line through A(–2, 3) and C(5, 3).

(c) Write the equation of the line through B and C.

Solution

(a) Since both A and B have an x-value of – 2, the equation is x = – 2.

The line is parallel to the y-axis.

(b) Since both A and C have a y-value of 3, the equation is y = 3.

The line is parallel to the x-axis.

(c) Here, we use the two point form to find the equation. The equation of BC is

23(x – 5) = 7(y – 3) becomes 23x – 7y – 94 = 0

Intercepts

Lines that cross the axes are said to have intercepts.

The point where a line crosses the x-axis is called the x-intercept or zero.

The point where a line crosses the y-axis is called the y-intercept or initial value.

Say we're asked to draw a graph of a line given its equation. Since it takes only two points to define a line, if we could find the x and y intercepts, we would only need to join them to draw the line. We could of course pick any two x-values, find the two corresponding y values, which would give us two points to join, but often, it's easier to just find the intercepts.

We know that the equation of the x-axis is y = 0 and the y-axis has equation x = 0, so, to find the intercepts of a line given its equation, we set y = 0 to find the x-intercept and set x = 0 to find the y-intercept.

Example

Find the x- and y-intercept of the line with equation 2x – 5y + 10 = 0.

Solution

Set y = 0 to get 2x – 0 + 10 = 0 or 2x = – 10 so x = – 5. The x-intercept is ( – 5, 0).

Set x = 0 to get 0 – 5y + 10 = 0 or – 5y = – 10 so y = 2. The y-intercept is (0, 2).

We can plot these two points, join them and have the graph of the line 2x – 5y + 10 = 0.

Using Intercepts to find Slope

It's pretty obvious from the diagram that once we've located the x- and y-intercepts of a line, we can determine its slope -- since we know the displacement ratio of rise to run to get from one point on the line to the other.

If ( a, 0 ) is the x-intercept and ( 0, b ) is the y-intercept of a line,

the slope of the line is

Example

Find the x-intercept, y-intercept, and the slope of the line with equation 5x + 3y – 9 = 0.

Set y = 0 for the x-intercept: 5x = 9

Set x = 0 for the y-intercept: 3y = 9 .

The x-intercept is the y-intercept is (0, 3).

The slope therefore is

Practice

1) List the x-intercept, y-intercept and slope of these lines:

a) 3x + 2y – 12 = 0	b) y = 4x – 16	(c) y = 7	(d) x = – 3
e) x – 3y + 7 = 0	f) 2x + 4y – 9 = 0	g) Ax + By + C = 0

2) Write the equation (in Ax + By + C = 0 form wherever possible) of the line through:

a) A( 2, 5) and B(2, – 17)

b) C(– 3, 4) and D(4, 4)

c) E(3, 0) and F(0, – 6)

Solutions

1) List the x-intercept, y-intercept and slope of these lines:

.

2) Write the equation (in Ax + By + C = 0 form wherever possible) of the line through:

.

( Analytic Geometry MathRoom Index )

(all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).