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 \[y - y_{1} = m(x - x_{1})\] where \[y_{1}\] is the coordinate of the Y-axis, m is the slope, and \[x_{1}\] is the coordinate on the X-axis.
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.
(Image will be Uploaded Soon)
To calculate the slope, the formula used is \[m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\].
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 \[x_{1}, y_{1} = (2, 5) and x_{2}, y_{2} =(6, 7) \]
Substituting the values into the formula,
\[m = \frac{7 - 5}{6 - 2}\]
\[m = \frac{2}{3}\]
Did You Know?
What happens if we interchange the values of \[x_{1}, y_{1} and x_{2}, y_{2}\]?
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 \[x_{1}, y_{1} and x_{2}, y_{2}\], we get \[x_{1}, y_{1} = (6,7) and x_{2}, y_{2} = (2,5)\].
\[m = \frac{5 - 7}{ 2 - 6}\]
\[m = \frac{-2}{-3} = \frac{2}{3}\]
Hence, any one of the two coordinates can be used as \[ x_{1}, y_{1} \] and the other as \[ x_{2}, y_{2} \].
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 \[x_{1}, y_{1} and x_{2}, y_{2}\], we get \[x_{1}, y_{1} = (2,5) and x_{2}, y_{2} = (6,7)\] and the slope is calculated as \[m = \frac{2}{3}\], substitute the value of m and any one point in the formula \[y - y_{1} = m(x - x_{1})\].
\[y - y_{1} = m(x - x_{1})\]
\[y - 5 = \frac{2}{3} (x - 2)\]
Cross-multiply and simplify:
\[y - 5 = \frac{2}{3} (x - 2)\]
\[ \Rightarrow 3 (y - 5) = 2 (x - 2)\]
\[ \Rightarrow 3y - 15 = 2x - 4\]
\[ \Rightarrow 3y - 2x = 15 - 4\]
\[ \Rightarrow 3y - 2x = 11\]
The same equation can be expressed in slope-intercept form by making the equations in terms of y as shown below.
\[ \Rightarrow 3y - 2x = 11\]
\[ \Rightarrow 3y = 2x + 11\]
\[ \Rightarrow y = \frac{2}{3}x + \frac{11}{3}\]
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 \[m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\].
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,
\[ \Rightarrow m = \frac{0 - 3}{-1 - 2}\].
\[ \Rightarrow m = \frac{-3}{-3}\].
\[ \Rightarrow m = 1 \].
Substitute the value of m and any coordinate into the formula \[y - y_{1} = m(x - x_{1})\].
\[y - y_{1} = m(x - x_{1})\]
\[y - 0 = 1(x - (-1)\]
Simplify the equations:
\[y - 0 = 1(x - (-1)\]
\[ \Rightarrow y = x + 1 \]
\[ \Rightarrow y - x = 1 \]
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 \[\frac{1}{3}\].
Substitute the value of m and the coordinate into the formula \[y - y_{1} = m(x - x_{1})\].
\[y - y_{1} = m(x - x_{1})\]
\[ \Rightarrow y - 3 = m(x - x_{1})\]
\[ \Rightarrow y - 3 = \frac{1}{3}(x - 1)\]
Cross multiply and simplify the equations:
\[ \Rightarrow y - 3 = \frac{1}{3}(x - 1)\]
\[ \Rightarrow 3(y - 3) = 1(x - 1)\]
Simplify the equations further:
\[ \Rightarrow 3(y - 3) = 1(x - 1)\]
\[ \Rightarrow 3y - 9 = x - 1\]
\[ \Rightarrow 3y - x = 8\]
The same equation can be expressed in slope-intercept form by making the equations in terms of y.
\[ \Rightarrow 3y - x = 8\]
\[ \Rightarrow 3y = x + 8\]
\[ \Rightarrow y = \frac{1}{3}x + \frac{8}{3} \]
The equation of the line passing through the point (1,3) and having a slope of \[\frac{1}{3}\] is \[ \Rightarrow 3y - x = 8 or \frac{1}{3}x + \frac{8}{3}\].
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. What is the equation of a line in coordinate geometry?
The equation of a line is an algebraic formula that represents all the points on a straight line in a coordinate plane. Any point lying on the line will satisfy its equation. The most common general form is Ax + By + C = 0, where A, B, and C are constants, and x and y are the coordinate variables. This allows us to algebraically analyse the line's properties, such as its slope and intercepts.
2. How do you find the equation of a line when two points are given?
To find the equation of a line passing through two points, (x₁, y₁) and (x₂, y₂), you can follow these steps:
- Step 1: Calculate the slope (m). Use the slope formula, m = (y₂ - y₁) / (x₂ - x₁).
- Step 2: Use the point-slope form. Substitute the calculated slope 'm' and the coordinates of one of the given points into the point-slope formula: y - y₁ = m(x - x₁).
- Step 3: Simplify the equation. Rearrange the equation into the standard form (Ax + By + C = 0) or the slope-intercept form (y = mx + c).
3. What are the three main forms for writing the equation of a straight line?
The equation of a straight line can be expressed in several forms, each highlighting different properties of the line. The three primary forms are:
- Slope-Intercept Form: y = mx + c. This form is useful because it directly gives the slope (m) and the y-intercept (c).
- Point-Slope Form: y - y₁ = m(x - x₁). This form is used when you know the slope (m) and the coordinates of a single point (x₁, y₁) on the line.
- Standard Form: Ax + By + C = 0. In this form, x and y are on the same side of the equation. It is a general form that can represent any straight line, including vertical lines.
4. Why is it impossible to determine a unique equation for a line if only one point is known?
A single point in a coordinate plane is not enough to define a unique line because an infinite number of different straight lines can pass through that one point, each with a different slope or orientation. To uniquely define a line, you need a second piece of information to constrain its direction, which can be either:
- A second point on the line.
- The slope of the line.
Without this second constraint, the line is not fixed in space.
5. When calculating the slope with two points, does it matter which point is chosen as (x₁, y₁)?
No, it does not matter. The value of the slope will be the same regardless of which point you designate as the first (x₁, y₁) or the second (x₂, y₂). The ratio of the difference in y-coordinates to the difference in x-coordinates remains constant. For example, if you swap the points, the signs of both the numerator (y₂ - y₁) and the denominator (x₂ - x₁) will flip, and the resulting negative signs will cancel each other out, yielding the same slope.
6. How do you find the slope of a line from its equation in standard form (Ax + By + C = 0)?
To find the slope from the standard form equation Ax + By + C = 0, you can rearrange it into the slope-intercept form (y = mx + c). By isolating 'y', the equation becomes By = -Ax - C, and then y = (-A/B)x - (C/B). From this, you can see that the coefficient of 'x' is the slope. Therefore, the slope (m) is m = -A/B.
7. How does the method for finding a line's equation in 3D space differ from 2D?
In 2D geometry, a line is defined by a point and a slope, which indicates its steepness. In 3D space, a simple slope is insufficient to describe a line's orientation. Instead, you need a point (x₁, y₁, z₁) on the line and a direction vector (a, b, c) that specifies the line's direction. The equation is then typically written in symmetric form as: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c. The direction vector serves the same purpose as the slope but accounts for all three dimensions.
