Lines Parallel to Axes
In two-dimensional geometry, there are two axes, which are the x-axis and the y-axis. A line that is parallel to the y-axis is of the form 'x=k', where 'k' is any real number and 'k' is the distance of the line from the y-axis. For example, the equation of a line which is of the form x = 3 is a line parallel to the y-axis and is 3 units away from the y-axis. Similarly, lines can be drawn parallel to the x-axis also. A line that is parallel to the x-axis is of the form 'y=k', where 'k' is a real number and is also the distance of the line from the x-axis. For example, the equation of a line which is of the form y = 2 is a line that is parallel to the x-axis and is 2 units away from the x-axis.
| 1. | Line Parallel to x-axis |
| 2. | Line Parallel to y-axis |
| 3. | Solved Examples |
| 4. | Practice Questions |
| 5. | FAQs on Lines Parallel to Axes |
Line Parallel to x-axis
A line that is parallel to the x-axis is of the form 'y = k', where 'k' is a constant value. In a coordinate plane, a straight line can be represented by an equation. To put the equation of this parallel line in a more generalized form, we can write it as 'y = k', where 'k' is any real number. Also, 'k' is said to be the distance from the x-axis to the line 'y=k'. For example, if the equation of a line is y = 5, then we can say that it is at a distance of 5 units above the x-axis line. All the points on a line that is parallel to the x-axis are at the same distance away from it.
Consider the equation y = 2, or y - 2 = 0. This is an equation with a single variable y. However, we can think of it as a two-variable linear equation in which the coefficient of x is 0:
0(x) + 1(y) + (-2) = 0.
Let us plot the graph for the equation, and find how the line 'y=2' will look.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
Substituting every value of 'x' given in the table, we see that the value of 'y' remains unchanged. For example, let us take the value of 'x = -4' and substitute in the equation, 0(x) + 1(y) + (-2) = 0.
0(-4) + 1(y) - 2 = 0
0 + y - 2 = 0
Therefore, y = 2.
Let us take a positive value for 'x = 3' and solve the equation to find the value of 'y'.
0(3) + 1 (y) - 2 = 0
0 + y - 2 = 0
y = 2.
Therefore, we can see that though the value of 'x' changes, the value of 'y' remains unchanged. Thus, all solutions of this linear equation are of the form (k,2), where k is some real number. The graph of the line 'y=2' is given below.
This is a line parallel to the x-axis. Thus, an equation of the form y = a represents a straight line parallel to the x-axis and intersecting the y-axis at (0,a).
Line Parallel to y-axis
A line that is parallel to the y-axis is x = k, where 'k' is a constant value. This means that for any value of 'y', the value of 'x' is the same. A more generalized way to represent an equation of a straight line parallel to the y-axis is x = k, where 'k' is a real number. Here, 'k' represents the distance from the y-axis to the line 'x=k'. For example, if we have the equation of a line as 'x =2', it says that the line is at a distance of 2 units away from the y-axis. All the points on a line that is parallel to the y-axis are at the same distance away from it.
Now, consider the equation x = 3. This can also be written as a two-variable linear equation, as follows:
1(x) + 0(y) + (-3) = 0.
Let us plot the graph for the equation, and find how the line 'x=3' will look.
| x | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| y | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
Substituting different values of 'y' in the equation, 1(x) + 0(y) + (-3) = 0, the value of 'x' remains unchanged. For example, if y = -3, then the value of 'x' is,
1(x) + 0(-3) +(-3) = 0.
x + 0 - 3 = 0
x -3 = 0
Therefore, x = 3.
Let us take a positive value for 'y'. Say 'y=2'. On substituting the value of 'y=2', we get,
1(x) + 0(2) + (-3) = 0
x + 0 -3 =0
Therefore, x = 3.
We can observe that for any value of 'y', the value of x = 3. Thus, the solutions of this equation are all of the form (3,k), where k is some real number. The graph of this equation will consist of all points whose x-coordinate is 3, that is, a line parallel to the y-axis, and passing through (3,0). The graph of the line whose equation is x = 3 is shown in the figure below.
In general, an equation of form x = a represents a straight line parallel to the y-axis and intersecting the x-axis at (a,0).
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Solved Examples
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Example 1: What does the equation 2x + 3 = - 1 represent when considered as a linear equation in two variables?
Solution: When considered as a linear equation in two variables, this represents a line parallel to the y-axis, as shown below.
Taking the equation 2x + 3 = -1, and solving for x, we get,
2x+3 = -1
2x = -1 -3
2x = -4
x = -4/2
x = -2
Therefore, the equation 2x + 3 = -1 represents a line that is parallel to the y-axis, which is x = -2. The line x = -2 is shown in the figure below.
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Example 2: The following figure shows four lines, each of which is parallel to one of the two axes. Determine the equation of each line.
Solution: \(L_{1}\) is parallel to the x-axis and passes through (0, 2). Thus, the equation of \({L_1}\) will be y = 2. \({L_2}\) is parallel to the y-axis and passes through (-1, 0). The equation of \({L_2}\) will be x = -1.
Similarly, the equation of \({L_3}\) will be \(y = - \frac{3}{2}\) and that of \({L_4}\) will be \(x = \frac{5}{2}\).
FAQs on Lines Parallel to Axes
What Does Parallel to the Axes Mean?
Parallel to axes means the lines that are parallel to either the x-axis or y-axis. A line parallel to the x-axis is a horizontal line whose equation is of the form y = k, where 'k' is the distance of the line from the x-axis. Similarly, a line parallel to the y-axis is a vertical line whose equation is of the form x = k, where 'k' is the distance of the line from the y-axis.
What is the Equation of the Line Parallel to x-axis?
The equation of the x-axis is given by y = 0. The equation of the line parallel to the x-axis is y = k, where 'k' is any real number. For example, considering the equation of a line, y = 2, for any value of 'x' the value of 'y' is always equal to 2. This can be understood by substituting various values of 'x' in the line equation, 0(x) + 1(y) - 2 = 0, which always results in y =2. This line is parallel to the x-axis.
What is the Equation of the Line Parallel to y-axis?
The equation of the y-axis is given by x = 0. The equation of the line parallel to the y-axis is x = k, where 'k' is any real number. For example, considering the equation of a line, x = 3, for any value of 'y' the value of 'x' is always equal to 3. This can be understood by substituting various values of 'y' in the line equation, 1(x) + 0(y) - 3 = 0, which always results in x = 3. This line is parallel to the y-axis.
When Can You Say That Two Lines are Parallel to the Axes?
All the vertical and horizontal lines on a plane are parallel to the axes. Horizontal lines are parallel to the x-axis while vertical lines are parallel to the y-axis. A line is parallel to axes if either the x-coordinate or y-coordinate is fixed or constant throughout the line and it should pass from either (0, a) or (a, 0). For example, a line with the equation, 3x - 6 = 0 is parallel to y-axis, since for any value of 'y' the value of x remains the same, which is 2. Similarly, the line with the equation 4y - 8 = 0 is parallel to the x-axis, since, for any value of 'x', the value of 'y' remains the same, which is 2.
What is the Equation of Line Parallel to y-axis and Passing Through (3, 4)?
The equation of the line parallel to the y-axis takes the form of x = k. The coordinate (3.4) lies on the equation of the line to be found. Therefore, substituting the value of 'x' in the equation 'x = k' , we get 3 = k or k = 3. Therefore the equation of the line parallel to the y-axis passing through (3.4) is 'x = 3'.