Lines Parallel to Axes
In twodimensional geometry, there are two axes, which are the xaxis and the yaxis. A line that is parallel to the yaxis is of the form 'x=k', where 'k' is any real number and 'k' is the distance of the line from the yaxis. For example, the equation of a line which is of the form x = 3 is a line parallel to the yaxis and is 3 units away from the yaxis. Similarly, lines can be drawn parallel to the xaxis also. A line that is parallel to the xaxis is of the form 'y=k', where 'k' is a real number and is also the distance of the line from the xaxis. For example, the equation of a line which is of the form y = 2 is a line that is parallel to the xaxis and is 2 units away from the xaxis.
1.  Line Parallel to xaxis 
2.  Line Parallel to yaxis 
3.  Solved Examples 
4.  Practice Questions 
5.  FAQs on Lines Parallel to Axes 
Line Parallel to xaxis
A line that is parallel to the xaxis 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 xaxis 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 xaxis line. All the points on a line that is parallel to the xaxis 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 twovariable 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 xaxis. Thus, an equation of the form y = a represents a straight line parallel to the xaxis and intersecting the yaxis at (0,a).
Line Parallel to yaxis
A line that is parallel to the yaxis 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 yaxis is x = k, where 'k' is a real number. Here, 'k' represents the distance from the yaxis 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 yaxis. All the points on a line that is parallel to the yaxis are at the same distance away from it.
Now, consider the equation x = 3. This can also be written as a twovariable 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 xcoordinate is 3, that is, a line parallel to the yaxis, 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 yaxis and intersecting the xaxis at (a,0).
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Solved Examples

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 yaxis, 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 yaxis, which is x = 2. The line x = 2 is shown in the figure below. 
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 xaxis and passes through (0, 2). Thus, the equation of \({L_1}\) will be y = 2. \({L_2}\) is parallel to the yaxis 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 xaxis or yaxis. A line parallel to the xaxis is a horizontal line whose equation is of the form y = k, where 'k' is the distance of the line from the xaxis. Similarly, a line parallel to the yaxis is a vertical line whose equation is of the form x = k, where 'k' is the distance of the line from the yaxis.
What is the Equation of the Line Parallel to xaxis?
The equation of the xaxis is given by y = 0. The equation of the line parallel to the xaxis 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 xaxis.
What is the Equation of the Line Parallel to yaxis?
The equation of the yaxis is given by x = 0. The equation of the line parallel to the yaxis 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 yaxis.
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 xaxis while vertical lines are parallel to the yaxis. A line is parallel to axes if either the xcoordinate or ycoordinate 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 yaxis, 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 xaxis, since, for any value of 'x', the value of 'y' remains the same, which is 2.
What is the Equation of Line Parallel to yaxis and Passing Through (3, 4)?
The equation of the line parallel to the yaxis 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 yaxis passing through (3.4) is 'x = 3'.
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