Equation of a Line: Definition, Different Equations, Examples

Equation of a Line: How far up a line is going from a point, A, to another one, B? How far along? What is the slope of a straight line or how steep is a line? While all these questions may not be as intimidating as they seem, sometimes students who are studying lines for the first time may have a hard time calculating distance. The reason is that they may not have any idea of what the equation of a line is, and what different forms of a line are. In this thorough article, we have explained the coordinate system in an easy to understand way. 

In algebra, an equation of a line is a form of representing a set of points. These points—in a 2D coordinate system—are also referred to as variables on an x and y-axis. There could also be a third axis if we are considering a 3D system. This set of variables, i.e., x and y, form an algebraic equation, known as ‘Equation of a line’. From Cartesian coordinate system to line formulae to straight line to line passing through a point to solved examples, this detailed guide focuses on everything related to lines. Read on to find out more.

Equation of a Line – Definition

Equation of a Line: The equation of a line in \(2\) dimensional coordinate system is an equation involving terms containing variables \(x\) and \(y\) separately with degree \(1\). Any equation with a power of exactly \(1\) is called linear. By ‘line’ here, we shall mean a straight line and hence, our discussion will be focused on the equation of a straight line in this topic.

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Cartesian Coordinate System

When two numbered lines, perpendicular to each other (usually vertical and horizontal) are placed together, so that their origins (the points corresponding to zero) coincide, then the resulting configuration is called a cartesian coordinate plane.

Coordinate of a Point

Let \(P\) be any point in the coordinate plane. From \(P\), draw \(PM\) perpendicular to \(X’OX\), then
(i) \(OM\) is called the \(x\)-coordinate (or abscissa) of \(P\) and is usually denoted by \(x\)
(ii) \(MP\) is called the \(y\)-coordinate (or ordinate) of \(P\) and is usually denoted by \(y\)
(iii) \(x\) and \(y\) taken together are called the cartesian coordinates or simply coordinates of the point \(P\) and are denoted by \((x, y)\).

Equation of a Line: Tl;dr

The equation of a line is the relation between the \(x\)-coordinate and \(y\)-coordinate of collection of points on the line.

How to Find Equation of a Line?

(i) Let \((x,y)\) be any point on the straight line.
(ii) Understand the geometrical condition governing the movement of this point \((x,y)\) on the line.
(iii) Express this condition in mathematical form in terms of \(x,y\) and known constant (or constants), if necessary.
(iv) The equation thus obtained will be the equation of the required straight line.

What is Equation of a Line Formula?

There are many formulas available for the equation of a straight line under different conditions.
They are:
a) general equation of a straight line.
b) equation of a line parallel to \(x-\) axis
c) equation of a line parallel to \(y-\) axis
d) equation of a line passing through a point
e) equation of a line when two points are given
f) equation of a line perpendicular to a line
g) equation of a line parallel to a line
h) equation of a line in point slope form
i) equation of a line in slope intercept form
j) equation of a line in intercept form
k) equation of a line in normal (or perpendicular) form
However, in this topic, we shall cover the equation of a straight line for the first six cases only

What is the General Equation of a Straight Line?

What is the Equation of a Line Parallel to \(x\)- axis?

Let \(AB\) be a straight line parallel to \(x\)- axis. Then, the ordinate of every point on the line \(AB\) is constant, say \(b\) (In the figure \(b\) is positive)
Let \(P(x,y)\) be any point on the line \(AB\). From \(P\), draw \(MP\) parallel on \(x\)-axis, then \(MP = y = b\) is always true.

Therefore, \(y = b\), which is the required equation of the line \(AB\).
If the straight line is parallel to \(x\)- axis, but towards the negative side of \(y\)- axis, then \(b\) will be negative.
Corollary: If \(b = 0\), then the line \(AB\) coincides with \(x\)-axis.
Hence, the equation of \(x\)-axis is \(y = 0\).

What is the Equation of a Line Parallel to \(y\)- axis?

Let \(AB\) be a straight line parallel to \(y-\) axis then the abscissa of every point on the line \(AB\) is constant, say \(a\).
Let \(P(x,y)\) be any point on the line \(AB\).
From \(P\), draw \(PN\) parallel \(x-\) axis, then \(NP = x = a\) is always true.
Therefore, \(x = a\) is the required equation of the line \(AB\).
If the straight line is parallel to \(y-\) axis, but towards the negative side of \(x-\) axis, then \(a\) will be negative.
Corollary: If \(a = 0\), then the line \(AB\) coincides with \(y-\) axis.
Hence, the equation of \(y-\) axis is \(x = 0\).

What is the Equation of a Line Passing Through a Point?

The equation of a straight line passing through a given point \((x_1,\,y_1 )\) is given by \((y – y_1 ) = m (x – x_1)\), where \((x,y)\) represent any point on the straight line and \(m\) is the gradient of the line given by \(m = \rm{tan} θ\), \(θ\) is the angle made by the line with the \(x-\)axis.

Practice Exam Questions

If the coordinates of two points are given as \((x_1 , y_1 )\) and \((x_2 , y_2 )\) then \(m\) is given by \(m = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}\).

What is the Equation of a Line when Two Points are Given?

The equation of a line passing through two points \((x_1 , y_1 )\) and \((x_2 , y_2 )\) is given by \(\frac{{y – {y_1}}}{{{y_2} – {y_1}}} = \frac{{x – {x_1}}}{{{x_2} – {x_1}}}\).

What is the Equation of a Line Perpendicular to a Line?

The equation of a line perpendicular to a given line \(ax + by + c = 0\) is \(bx – ay + d = 0\), where \(d\) is a constant.
So, in order to obtain the equation of a straight line perpendicular to a given line, interchange the coefficients of \(x\) and \(y\), multiply the new coefficient of \(y\) of the perpendicular line with \(-1\). The constant \(c\) will change to a new constant \(d\).

As we mentioned earlier, there are many other forms of equations of a straight line that are not discussed here.

Solved Examples – Equation of a Line

Q.1. If the line passes through the points \((6,-2)\) & \((2,10)\). Find its slope.
Ans: Consider \((x_1 , y_1 )=(6 , -2)\) and \((x_2 , y_2 ) = (2 , 10)\)
Slope of an equation is given by \(m = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}\)
\( \Rightarrow m = \frac{{10 – ( – 2)}}{{2 – 6}} = \frac{{12}}{{ – 4}} = – 3\)
Therefore, the slope of the equation is \(-3.\)

Q.2. Write an equation of a line that is parallel to the \(x-\)axis at a distance of \(7\) units at the positive direction of it.
Ans: We know that the equation of a line parallel to the \(x-\)axis is \(y = b\), where \(b\) is the distance of the line from it.
Here \(b = 7\) units at the positive direction of \(y-\)axis.
Hence, the required equation is \(y = 7\).

Q.3. Find the equation of a line that passes through the points \((-1, 0)\) & \((- 4, 12)\).
Ans: From the given \((x_1, y_1 ) = (- 1, 0)\) and \((x_2, y_2 ) = (- 4, 12)\)
Hence, the equation of the line is \(\frac{{y – {y_1}}}{{{y_2} – {y_1}}} = \frac{{x – {x_1}}}{{{x_2} – {x_1}}}\)
\( \Rightarrow \frac{{y – 0}}{{12 – 0}} = \frac{{x – ( – 1)}}{{ – 4 – ( – 1)}}\)
\( \Rightarrow \frac{y}{{12}} = \frac{{x + 1}}{{ – 4 + 1}}\)
\( \Rightarrow \frac{y}{{12}} = \frac{{x + 1}}{{ – 3}}\)
\( \Rightarrow \frac{y}{4} = \frac{{x + 1}}{{ – 1}}\)
\( \Rightarrow 4x + 4 = – y\)
\( \Rightarrow 4x + y + 4 = 0\)
Therefore, the obtained equation of a line is \( 4x + y + 4 = 0\)

Q.4. What is the equation of the line through the points \((-3, 0)\) & \((-3, 5)\).
Ans: The \(x\)-coordinate of both the points are the same. This means that both the points are at a distance of \(3\) units from the \(y-\)axis at the negative side of \(x-\)axis and hence, the line is parallel to the \(y-\)axis.
So, the required equation of the line is \(x = – 3\).

Q.5. What is the equation of the line through the points \((-3, 6)\) & \((1, 6)\).
Ans: The \(y-\) coordinate of both the points are the same. This means that both the points are at a distance of \(6\) units from the \(x-\)axis at the positive side of \(y-\)axis and hence, the line is parallel to the \(x-\)axis.
So, the required equation of the line is \(y = 6\).

Q.6. What is the equation of the line through the points \((-6, 0)\) & \((-6, 5)\).
Ans: The \(x-\)coordinate of both the points are the same. This means that both the points are at a distance of \(6\) units from the \(y-\)axis at the negative side of \(x-\)axis and hence, the line is parallel to the \(y-\)axis.
So, the required equation of the line is \(x = – 6\).

Summary

The equation of a straight line is available in many forms. However, only a few forms like the general equation of a straight line, equation of a line parallel to \(x\) and \(y-\) axis, equation of a line passing through one and two points, and equation of a line perpendicular to a line are described here. This topic will lay a foundation for the students to understand the equation of a straight line.

FAQs on Equation of a Line

Here are some of the FAQs about the equation of a Line:

Q.1. What is the straight-line formula?
Ans: The equation of a straight line is available in many forms depending on what information is available.
But, the general form of the equation of a straight line is given by \(Ax + By = C\), where \(A, B\) and \(C\) are constants (not all zero) and \((x, y)\) represent any point on the straight line.

Q.2. How do you find the equation of a line?
Ans: Follow the steps below to find the equation of a line:
(i) Understand the geometrical condition governing the movement of any point \((x, y)\) on the line.
(ii) Express this condition in mathematical form in terms of \(x, y\) and known constant (or constants), if necessary.
(iii) The equation thus obtained will be the equation of the required straight line.

Q.3. How do you find the equation of a line given two points?
Ans: The equation of a line when the two points are given as \((x_1, y_1 )\) and \((x_2, y_2 )\) is given by \(\frac{{y – {y_1}}}{{{y_2} – {y_1}}} = \frac{{x – {x_1}}}{{{x_2} – {x_1}}}\)

Q.4. What is the equation for a vertical line?
Ans: The equation of a vertical line is \(x = ± a\), where \(a\) is the distance of the line from \(y-\)axis.

Q.5. How do you graph a line from an equation?
Ans: The equation of a straight line is available in many forms depending on what information is available. A graph can be drawn from any form of the equation of a straight line.
For example if the equation is given in slope-intercept form \(y = mx + c\). Then to draw the graph:
a) Take \((0, c)\) as one point.
b) Get another point by using the slope \(m\) with \(y-\) intercept at the reference point.
c) Joins the two points.

Q.6. Define Equation of a Line?
Ans: The equation of a line is the relation between the \(x-\) coordinate and \(y-\)coordinate of any point on the line.

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