Equations of Lines Parallel to Y-axis and Y-axis: Linear equations describe the physical phenomenon around us, such as the speed of an object, the speed of a train that runs between two stations, etc. In Mathematics, linear equations can be drawn on graph paper, and they look like a straight line. The branch of Mathematics that discusses graphs is coordinate geometry. In this article, we will look into detail about the equation of lines parallel to the X-axis and Y-axis

In coordinate geometry, there are mainly two axes: the \(x\)-axis and the \(y\)-axis. If we draw the lines in the graph, sometimes they pass through the origin, some intersect the axes, some parallel to axes, etc. The equation of the line that is parallel to the \(y\)-axis is in the form \(x=k\). The equation of the line that is parallel to the \(y\)-axis is in the form \(y=k\).

Coordinate Axes

Coordinate geometry is the branch of mathematics that deals with algebra and geometry. It is used to study geometric figures like parabolas and straight lines by using numbers.

There are mainly two axes in the geometry. One is horizontal (\(x\)-axis), and the other is vertical (\(y\)-axis). These two axes meet at the intersection point known as the origin.

Learn About the Equation of a Line

The location of the point can be shown by using an ordered pair \((x, y)\). Here

1. \(x\)-abscissa
2. \(y\)-ordinate

On the right side of the \(x\)-axis we have positive values, and on the left side, we have negative values. Similarly, on the upside of the \(y\)-axis, the values are positive, and bottom side, the values are negative.

Equations of Lines Parallel to X-axis and Y-axis:

A straight line has infinitely many solutions. The straight-line graph is obtained by joining some of the points (solutions) of the line in a cartesian plane. The graph of the line depends on the position of the point that represents the solution.

Sometimes the graph of the line passes through the origin, and sometimes it intersects the axes: \(x\)-axis, \(y\)-axis. And, sometimes, it is parallel to the axes: \(x\)-axis and the \(y\)-axis. The general form of the line parallel to \(x\)-axis for any real number \(k\) is in the form of \(y=k\) And in the equation \(y=k\), the real number \(k\) gives the distance of the point (solution of the line) from the \(x\)-axis.

For example, the equation of the line, that is in the form of \(y=5\), is a line parallel to the \(x\)-axis and passing through the point \((0,5)\) on the \(y\)-axis. Here, the graph of the given linear equation \(y=5\) describes that it is at a distance of \(5\) units from the \(x\)-axis.

The general form of the line parallel to \(y-\) axis for any real number \(k\) is in the form \(x=k\). And in the equation \(x=k\), the real number \(k\) gives the distance of the point (solution of the line) from the \(y\)-axis.

For example, the equation of the line, that is in the form of \(x=5\), is a line parallel to the \(y\)-axis and passing through the point \((5,0)\) on the \(x\)-axis. Here, the graph of the given linear equation \(x=5\) describes that it is at a distance of \(5\) units from the \(y\)-axis.

Equation of a Line Parallel to X-axis

We can write the equation of the line parallel to \(x\)-axis in generalised form as in the form of \(y=k(k \in R)\), where \(k\) be any real-valued number.

Here, in the equation of the line parallel to \(x\)-axis, which is in the form of \(y=k\), the real number \(k\) gives the distance of the line from the \(x\)-axis. The line parallel to the \(x\)-axis lies either above or below the \(x\)-axis.

1. The equation of the line \(y=k\), lies parallel to \(x\)-axis at a \(k\) units distance from above the \(x\)-axis.
2. The equation of the line \(y=-k\), lies parallel to \(x\)-axis at a \(k\) units distance from below the \(x\)-axis.

All the points that lie on the straight line of the equation \(y=k\) lie at the same distance from the \(x\)-axis. For example, consider the equation of the line in the form of \(y=2\). The above equation \((y=2)\) is the equation in a single variable. We can write the above equation in the form of \(y-2=0\) or \(0 \times x+y-2=0\).

\(x\)	\(0\)	\(1\)	\(2\)	\(-1\)	\(-2\)
\(y\)	\(2\)	\(2\)	\(2\)	\(2\)	\(2\)

From the table, we can see that the value of \(y\) is constant \((2)\) for all values of the variable \(x\). Plot the points and draw the line connecting the points, which gives a parallel line to the \(x\)-axis.

Thus, the line \(y=2\) is drawn in the cartesian plane is parallel to the \(x\)-axis and lies \(2\) units above the \(x\)-axis.

Equation of a Line Parallel to Y-axis

We can write the equation of the line parallel to \(y\)-axis in generalised form as in the form of \(x=c(c \in R)\), where \(c\) be any real-valued number.

Here, in the equation of the line parallel to \(y\)-axis, which is in the form of \(x=c\), the real number \(c\) gives the distance of the line from the \(y\)-axis. The line parallel to \(y\)-axis lies either to the right of the \(x\)-axis or to the left of the \(y\)-axis.

1. The equation of the line \(x=c\), lies parallel to \(y\)-axis at \(c\) units at a distance from the right of the \(y\)-axis.
2. The equation of the line \(x=-c\), lies parallel to \(x\)-axis at \(c\) units at a distance from the left of the \(y\)-axis.

All the points that lie on the straight line of the equation \(x=c\) lie at the same distance from the \(y\)-axis. For example, consider the equation of the line in the form of \(x=3\). The above equation \((x=3)\) is the equation in a single variable. We can write the above equation in the form of \(x-3=0\) or \(x+0 \times y-3=0\).

\(x\)	\(3\)	\(3\)	\(3\)	\(3\)	\(3\)
\(y\)	\(0\)	\(1\)	\(2\)	\(-1\)	\(-2\)

From the table, we can see that the value of \(x\) is constant \((3)\) for all values of the variable \(y\). Plot the points and draw the line connecting the points, which gives the parallel line to the \(y\)-axis. Thus, the line \(x=3\) is drawn in the cartesian plane is parallel to the \(y\)-axis and lies \(3\) units to the right of the \(y\)-axis.

Solved Examples – Equations of Lines Parallel to Y-axis and Y-axis

Q.1. What does the graph of the linear equation \(2 x+3=-1\) represent in the cartesian plane?
Ans:
Given linear equation is \(2 x+3=-1\)
Solving for the \(x\), we get,
\(\Rightarrow 2 x=-1-3\)
\(\Rightarrow 2 x=-4\)
\(\Rightarrow x=-\frac{4}{2}\)
\(\Rightarrow x=-2\)
This indicates that for all values of \(y\), the value of the variable \(x=-2\), which is constant. It indicates that the graph of the line \(x=-2\) is the line parallel to the \(y\)-axis and at a distance of \(2\) units to the left of \(y\)-axis.

Q.2. Draw the line graph, which is parallel to \(x\)-axis and at a distance of \(2\) units above the \(x\)-axis.
Ans:
We know that a line parallel to \(x\)-axis and at a distance of \(k\) units above the \(x\)-axis is in the form of \(y=k\)
Given, the line is parallel to \(x\)-axis and at a distance of \(2\) units above the \(x\)-axis.
The equation of the line is in the form of \(y=2\).
The graph of the equation \(y=2\) is given below:

Q.3. The four lines are shown in the figure below, parallel to any one of the axes. Observe the lines and write the equations of the line.

Ans:
In the given figure, the lines \(L_{1}\) and \(L_{3}\) are parallel to \(x\)-axis. And, the lines \(L_{2}\) and \(L_{4}\) are parallel to \(y\)-axis.
We know that a line parallel to \(x\)-axis and at a distance of \(k\) units above the \(x\)-axis is in the form of \(y=k\), where \(k\) is a constant number, and it is the value where the line meets the \(y\)-axis.
The line \(L_{1}\) intersects \(y\)-axis at \(2\) and the line \(L_{3}\) intersects \(y\)-axis at \(-1.5=-\frac{3}{2}\).
Therefore, the equations of the lines \(L_{1}\) and \(L_{3}\) are \(L_{1}: y=2\) and \(L_{3}: y=-\frac{3}{2}\)
We know that a line parallel to \(y\)-axis and at a distance of \(k\) units right of the \(y\)-axis is in the form of \(x=k\), where \(k\) is a constant number, and it is the value where the line meets the \(x\)-axis.
The line \(L_{2}\) intersects \(x\)-axis at \(-1\) and the line \(L_{4}\) intersects \(x\)-axis at \(2.5=\frac{5}{2}\).
Therefore, the equations of the lines \(L_{2}\) and \(L_{4}\) are \(L_{2}: x=-1\) and \(L_{4}: x=\frac{5}{2}\).

Q.4. Check the line \(x=-7\) is parallel to \(y\)-axis or not? If it is parallel, draw the graph.
Ans:
Given equation is \(x=-7\).
We know that equation \(x=k\) is parallel to \(y\)-axis.
So, the given line \(x=-7\) is parallel to \(y\)-axis and at a distance of \(7\) units to the left of \(y\)-axis.

Q.5. The line \(2 y+6=0\) is parallel to which axes?
Ans:
Given line is \(2 y+6=0\)
\(\Rightarrow y=-\frac{6}{2}=-3\)
The given line is in the form of \(y=k\), and we know that a line in the form of \(y=k\) lies parallel to \(x\)-axis.
Therefore, the given line \(2 y+6=0\) is parallel to \(x\)-axis.

Summary

In this article, we have discussed the cartesian plane and graph of the linear equations. This article also discussed the graphs of equations parallel to axes. We studied the equations and graphs of the equations parallel to \(x\)-axis or \(y\)-axis with the help of solved examples.

Learn About Angle Between Two Lines

Frequently Asked Questions

Q.1. What is the equation of a line which is parallel to the x-axis and is at a distance of 6 units above?
Ans: The equation of the line parallel to \(x\)-axis and at a distance of 6 units above is given by \(y=6\) or \(y-6=0\)

Q.2. What equation is parallel to the y axis?
Ans: The equation parallel to \(y\)-axis in the form of \(x=k\).

Q.3. What equation is parallel to the x-axis?
Ans: The equation parallel to \(x\)-axis in the form of \(y=k\).

Q.4. Give an example of equations of lines parallel to the x-axis and y-axis?
Ans: The examples of equations of lines parallel to \(x\)-axis and \(y\)-axis are \(y=2, x=3\) etc.

Q.5. Is \(x=-2\) is parallel to y-axis?
Ans: Yes. The given line \(x=-2\) is parallel to \(y\)-axis.

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