How to Find the Equation of a Line Using Slope Intercept and Point Slope Formula
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.
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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:
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 of a Line in Coordinate Geometry
1. What is the equation of a line?
The equation of a line is a mathematical statement that represents all the points lying on a straight line in a coordinate plane. In two-dimensional geometry, it is commonly written in the form y = mx + c, where:
- m is the slope (gradient) of the line
- c is the y-intercept (where the line crosses the y-axis)
2. What is the slope-intercept form of a line?
The slope-intercept form of a line is y = mx + c, where m is the slope and c is the y-intercept. This form makes it easy to:
- Identify the slope (m)
- Locate the y-intercept (c)
- Graph the line quickly
3. How do you find the equation of a line given two points?
To find the equation of a line given two points, first calculate the slope and then use the point-slope form. Steps:
- Use the slope formula: m = (y₂ − y₁)/(x₂ − x₁)
- Substitute into point-slope form: y − y₁ = m(x − x₁)
- Simplify to slope-intercept form if needed
- m = (6 − 2)/(3 − 1) = 4/2 = 2
- Equation: y − 2 = 2(x − 1)
- Simplified: y = 2x
4. What is the point-slope form of a line?
The point-slope form of a line is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a known point on the line. This form is useful when:
- You know the slope
- You are given one point
5. What is the general form of a linear equation?
The general form of a linear equation is Ax + By + C = 0, where A, B, and C are constants and A and B are not both zero. This form is useful because:
- It works for all straight lines, including vertical lines
- It is commonly used in higher algebra and coordinate geometry
6. How do you find the slope of a line?
The slope of a line measures its steepness and is calculated using m = (y₂ − y₁)/(x₂ − x₁). Steps:
- Choose two points (x₁, y₁) and (x₂, y₂)
- Subtract the y-values
- Subtract the x-values
- Divide the differences
7. What is the equation of a horizontal line?
The equation of a horizontal line is y = constant. This means:
- The slope is 0
- The y-value stays the same for all x-values
8. What is the equation of a vertical line?
The equation of a vertical line is x = constant. This means:
- The slope is undefined
- The x-value remains fixed while y changes
9. How do you write the equation of a line parallel to another line?
To write the equation of a line parallel to another line, use the same slope as the given line. Steps:
- Identify the slope m of the original line
- Use point-slope form with the new point
- y − 5 = 3(x − 2)
- Simplified: y = 3x − 1
10. What is the difference between slope-intercept form and standard form?
The main difference is that slope-intercept form shows the slope directly, while standard (general) form does not.
- Slope-intercept form: y = mx + c (easy to graph using slope and intercept)
- Standard/general form: Ax + By + C = 0 (more suitable for algebraic manipulation)
