Ninth Grade Mathematics: Equations of Lines Parallel to Axes (x=k, y=k)
Most straight lines on the Cartesian plane cut across both coordinate axes diagonally. However, lines that run parallel to one of the axes represent special geometric cases where one of the variables is completely constant regardless of the value taken by the other variable. In this chapter, you will explore the algebraic equations and geometric properties of horizontal and vertical lines.
Equations of Lines Parallel to the x-Axis (Horizontal Lines)
A horizontal line runs parallel to the horizontal x-axis and perpendicular to the vertical y-axis. Every single point lying on a horizontal line shares the exact same y-coordinate, regardless of its x-position.
General Equation of a Horizontal Line:
y = k (or 0*x + 1*y - k = 0)
where k is a fixed real constant.
y
|
3 +--------------------------- y = 3
|
2 +
|
1 +
|
---+---+---+---+---+---+---+---+---+---+--- x
-3 -2 -1 0 1 2 3 4
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-1 +
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-2 +--------------------------- y = -2
|
In the line y = 3, points such as (-2, 3), (0, 3), (1, 3), and (5, 3) all satisfy the equation. The value of x can be anything, but y is locked at 3. The slope of any horizontal line is exactly zero (m = 0) because there is zero vertical change (rise = 0).
The Equation of the x-Axis Itself
The x-axis is itself a horizontal line that passes through the origin (0, 0). Because every point on the x-axis has a y-coordinate of zero, the algebraic equation of the x-axis is:
Equation of the x-Axis:
y = 0
Equations of Lines Parallel to the y-Axis (Vertical Lines)
A vertical line runs parallel to the vertical y-axis and perpendicular to the horizontal x-axis. Every point on a vertical line shares the exact same x-coordinate, regardless of its y-position.
General Equation of a Vertical Line:
x = k (or 1*x + 0*y - k = 0)
where k is a fixed real constant.
x = -2 x = 3
| |
| y |
| | |
| 2 + |
| | |
| 1 + |
| | |
-----------------+---+---------+-------- x
-2 0 3
| | |
|-1 + |
| | |
For the line x = 3, points such as (3, -4), (3, 0), (3, 2), and (3, 100) all lie on the line. The x-coordinate is locked at 3, while y can take any real value. The slope of a vertical line is undefined because the horizontal run is zero, and division by zero is impossible.
The Equation of the y-Axis Itself
The y-axis is itself a vertical line passing through the origin. Since every point on the vertical y-axis has an x-coordinate of zero, the algebraic equation of the y-axis is:
Equation of the y-Axis:
x = 0
Summary Comparison of Axis-Parallel Lines
Understanding the symmetry between horizontal and vertical lines is essential:
Property Horizontal Line (y = k) Vertical Line (x = k)
---------------------------------------------------------------------------------
Parallel to x-axis y-axis
Perpendicular to y-axis x-axis
Intercept y-intercept at (0, k) x-intercept at (k, 0)
Slope m = 0 Undefined
Standard Form 0*x + 1*y - k = 0 1*x + 0*y - k = 0
Fixed Coordinate y is constant x is constant
Chapter Exercises
Exercise 1: Write the equations of the lines passing through the point (-4, 7) that are: (a) parallel to the x-axis, and (b) parallel to the y-axis.
Exercise 2: Identify whether each of the following equations represents a horizontal line, a vertical line, or an oblique (slanted) line, and state its slope: 3x - 9 = 0, 4y + 12 = 0, 2x + 5y = 10, and y = 0.
Exercise 3: Find the point of intersection between the line x = -5 and the line y = 8. Draw an ASCII sketch verifying their intersection.
Solutions and Explanations
Solution 1: The given point has coordinates x = -4 and y = 7. A line parallel to the x-axis is horizontal and has the form y = k. Since it passes through (-4, 7), its y-coordinate is fixed at 7, yielding the equation y = 7 (or 0x + y - 7 = 0). A line parallel to the y-axis is vertical and has the form x = k. Since it passes through (-4, 7), its x-coordinate is fixed at -4, yielding the equation x = -4 (or x + 0y + 4 = 0).
Solution 2: For 3x - 9 = 0, solving gives 3x = 9, so x = 3. This equation contains no y-variable and represents a vertical line parallel to the y-axis; its slope is undefined. For 4y + 12 = 0, solving gives 4y = -12, so y = -3. This contains no x-variable and represents a horizontal line parallel to the x-axis; its slope is 0. For 2x + 5y = 10, both variables have non-zero coefficients; solving for y gives y = (-2/5)*x + 2, which is an oblique (slanted) line with slope m = -2/5. For y = 0, this is the equation of the x-axis itself, representing a horizontal line with slope m = 0.
Solution 3: The line x = -5 consists of all points whose x-coordinate is -5. The line y = 8 consists of all points whose y-coordinate is 8. The unique point that satisfies both conditions simultaneously is (-5, 8).
y
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(-5, 8) |
+----------+---------------- y = 8
| |
| |
--------+----------+---------------- x
-5 0
|
x = -5
The vertical line and horizontal line intersect at a right angle (90 degrees) at the coordinates (-5, 8).