

Equation of a Line Passing through Two Points
The equation of a line is an algebraic method to represent a set of points that together form a line in a coordinate system. The various points that together form a line in the coordinate axis can be represented as a set of variables (x, y) in order to form an algebraic equation, also referred to as the equation of a line. By using the equation of a line, it is possible to find whether a given point lies on the line.
The equation of any line is a linear equation having a degree of one. Let us read through the entire article to understand more about the different forms of an equation of a line and how we can determine the equation of a line.
A line segment can be defined as a connection between two points. Any two points, in two-dimensional geometry, can be connected using a line segment or simply, a straight line. The equation of a line can be found in the following three ways.
Slope Intercept Method
Point Slope Method
Standard Method
When two points that lie on a particular line are given, usually, the point-slope method is followed.
The equation of a line is
Finding the Slope of the Line Passing through Two Given Points
The slope or gradient of a line is the changing height of the line from the X-axis. For every unit of X, a change in Y on the line is known as the slope of a line.
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To calculate the slope, the formula used is
Here, the points are (2,5) and (6,7).
So, comparing the point to the general notation of coordinates on a Cartesian plane, i.e., (x, y), we get
Substituting the values into the formula,
Did You Know?
What happens if we interchange the values of
The value of m remains unchanged. The positioning of the coordinates does not affect the value of the slope.
Taking the same example as above but interchanging the values of
Hence, any one of the two coordinates can be used as
Finding the Equation of the Line Passing through Two Given Points
Steps to find the equation of a line passing through two given points is as follows:
Find the slope/gradient of the line.
Substitute the values of the slope and any one of the given points into the formula.
Simplify to obtain an equation resembling the standard equation of the line, i.e., Ax + By + C = 0, where A, B, and C are constants.
Taking the above example, where
Cross-multiply and simplify:
The same equation can be expressed in slope-intercept form by making the equations in terms of y as shown below.
Solved Examples
1. Find the equation of the line passing through the points (2,3) and (-1,0).
For calculating the slope, the formula used is
Here, the points are (2,3) and (-1,0)
So, comparing the point to the general notation of coordinates on a Cartesian plane, i.e., (x, y), we get (x1,y1) = (2,3) and (x2,y2) = (-1,0).
Substituting the values into the formula,
Substitute the value of m and any coordinate into the formula
Simplify the equations:
The same equation can be expressed in slope-intercept form by making the equations in terms of y.
y = x + 1
The equation of the line passing through the points (2,3) and (-1,0) is y = x + 1 or y - x = 1.
2. Find the Equation of the Line Passing through the Point (1,3) and Having a Slope
Substitute the value of m and the coordinate into the formula
Cross multiply and simplify the equations:
Simplify the equations further:
The same equation can be expressed in slope-intercept form by making the equations in terms of y.
The equation of the line passing through the point (1,3) and having a slope of
Conclusion
The equation of a line can be easily understood as a single representation for numerous points on the same line. The equation of a line has a general form, that is, ax + by + c = 0, and it must be noted that any point on this line satisfies this equation. There are two absolutely necessary requirements for forming the equation of a line, which are the slope of the line and any point on the line.
FAQs on Equation Line
1. Can we find the equation of a line, if only one coordinate is given? If not, what additional inputs are required to find the equation of the line passing through the given point?
Finding the equation of the line when only one coordinate is given is not possible because when a point is existing in space, an infinite number of lines can be passing through it. So, finding the one particular equation will be like finding a needle in a haystack.
For finding the correct or desired equation, we must have either the slope of the line or the second set of coordinates. The slope will help us get an idea about the height of the line and the other coordinate can give an estimate about the length of the line.
Moreover, as the formula to calculate the equation of the line is known,
2. How can we find the equation of a line passing through two points in 3D?
Like in the two-dimensional plane, we need a slope and a point through which the line passes, in a three-dimensional plane, a point through which the line passes is needed, along with a direction vector to entail the direction of the line.
In two-dimensional geometry, the slope gives the depth or height of the line. Similarly, in three-dimensional geometry, the idea of the direction of the line whose equation has to be derived is given by the direction vector.
The formula to find the equation passing through two points in 3d is
3. How can we determine the slope using the equation of a line?
If a line has an equation ax + by + c = 0, then its slope will be

















