Equation of a line from Point-slope form

By Veronica Cruz

The point-slope form is a linear equation used to represent the line. This form consists of the coordinates of a single point on the line and the slope of that line. The point-slope formula is given as 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1), where (𝑥1, 𝑦1) represents the point, and m represents the slope of the line. By having the slope and the point information, it becomes very straightforward to plot the line and determine the other characteristics of the line.  

The point-slope form is particularly useful when we know a point on the line and the slope of the line. Using this equation, we can easily calculate the equation of the line without knowing the y-intercept. Also, the point-slope form helps to solve real-life problems by finding the equation of the line passing through two given points. Moreover, the Point-slope form of the equation is noticeably clear and concise; it does not represent anything extra compared to the other forms.  

Derivation of point-slope form  

The derivation of the point-slope form is quite simple and can be understood easily by considering two points on a line.  

Consider a line that passes through two points, P1 (𝑥1, 𝑦1) and P2 (𝑥2, 𝑦2). The slope of this line can be calculated using the formula (𝑦2 − 𝑦1)÷ (𝑥2 − 𝑥1). This formula gives us a rate of change of the line, which is the difference between the x and y coordinates of the two points.  

Now we have the slope of the line, m, and we can write the line equation using the slope-intercept form, y = mx + b, where b is the y-intercept. We know that any point on the line must satisfy this equation, so P1 (𝑥1, 𝑦1) must satisfy the equation 𝑦1 = 𝑚𝑥1 + 𝑏.

Solving for b, we get [Equation].  

Substituting b in the slope-intercept form, we get [Equation].   

Simplifying this expression, we get [Equation], the point-slope form of the Equation of a line.  

How to find the equation of a line using point-slope form?  

The equation of a line in point-slope form is:  

[Equation]  

where:  

  • ([Equation]) is a point on the line  
  • m is the slope of the line  

To use the point-slope form to find the equation of a line, you need to know at least one point on the line and its slope. Here is how you can use the point-slope form:  

  1. Identify a point (x1, y1) on the line.  
  1. Determine the slope (m) of the line. This can be done by using the coordinates of two points on the line or other given information, such as the angle of inclination or the equation of a perpendicular or parallel line.  
  1. Substitute the values of x1, y1, and m into the point-slope form equation.  
  1. Simplify and rearrange the equation to put it into the desired form, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C).  

Here are three examples of finding the equation of a line using point-slope form:  

  • Example 1:

Find the Equation of the line that passes through the point (2, 3) with a slope of -4.  

Solution: Using the point-slope form, we have: y – 3 = -4(x – 2)   

Simplifying, we get: y – 3 = -4x + 8 y = -4x + 11  

Therefore, the equation of the line is y = -4x + 11.  

  • Example 2:

Find the Equation of the line that passes through the point (5, -1) with a slope of 2/3.  

Solution: Using the point-slope form, we have: y – (-1) = 2/3(x – 5)   

Simplifying, we get: y + 1 = 2/3x – 10/3 y = 2/3x – 13/3  

Therefore, the equation of the line is y = 2/3x – 13/3.  

  • Example 3:

Find the Equation of the line that passes through the point (-2, 7) and is parallel to the line y = 3x + 2.  

Solution: Since the given line has a slope of 3, any parallel line will also have a slope of 3. Using the point-slope form, we have: y – 7 = 3(x – (-2))   

Simplifying, we get: y – 7 = 3x + 6 y = 3x + 13  

Therefore, the equation of the line is y = 3x + 13.  

Applications of point-slope form  

The point-slope form of a linear equation is useful in various applications in mathematics and science. Here are a few illustrations of how it can be employed:  

  • Finding the equation of a line: The point-slope form is particularly useful when you know the slope of a line and one point on that line because it provides a direct way to write the equation of the line.  
  • Graphing linear functions: The point-slope form is useful when graphing linear functions because it provides a starting point for the graph and can be easily used to find other points on the line.  
  • Calculating rates of change: The slope in the point-slope form represents the rate of change between two points on the line. For example, if you have a line representing the distance an object travels over time, the slope of the line can be used to calculate the object’s speed.  
  • Solving real-world problems: The point-slope form can be used to model real-world situations that involve linear relationships, such as growth rates or depreciation of assets over time.  
  • Calculating perpendicular lines: The point-slope form can be used to find the equation of a line perpendicular to another line. If you know the slope of the first line, you can find the negative reciprocal of that slope to find the slope of the perpendicular line. Then, using the point-slope form, you can find the equation of the perpendicular line that passes through a given point.  

In conclusion, a point-slope form of the equation is an essential tool in algebraic mathematics. It is easy to use and understand and can be extremely helpful when determining the equation of the line with given information about the slope and point. Furthermore, this form of a linear equation can be useful for solving real-life problems and graphing lines. Using point-slope form, we can quickly and straightforwardly assign different qualities of the line without any mathematical intricacy. 

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