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.
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Slope Intercept Method
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Point Slope Method
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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.
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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:
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Find the slope/gradient of the line.
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Substitute the values of the slope and any one of the given points into the formula.
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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.