What is the Point-Slope Form and what other methods are present to find the equation of the line?
Point slope form is one of the types used to write the equation of the line. The other methods are two-point foam, slope-intercept form, and intercept form. An equation that satisfied a line on every single point of the line is said to be the equation of the line.
In this post, we shall discuss the different types to write the equation of the line with a lot of examples.
What is the point-slope form?
Point slope form is a type of the equation of the line. It is generally that type of equation of line used to represent a straight line with the help of points and slope of the given line. The slope is a mathematical term used to tell the steepness of the line.
It is clear by the name of the point-slope form that it must be dependent on the points and the slope of the line. In this type an equation of the line is formed by the slope, generally represented by m, passing through the points (x1, y1).
This type of equation of the line is generally written as.
y – y1 = m (x – x1)
x and y in this above equation are random variables, x1 and y1 are fixed points from which the line passes, and m is the slope of the line.
The above-mentioned formula or equation is used to calculate the equation of the line. This type of equation of line follows slope and the points for the calculation to represent an equation of the line. The above-mentioned equation can only be applicable to find the equation of a line when the slope m and the points (x1, y1) are known.
How to calculate the equation of the line by using a point-slope form?
Point slope form calculator is an online tool that finds the point-slope form using a point and slope of the line.
For manual calculation of the equation of the line by using the point-slope form, keep in mind that first of all identify the slope m and the fixed points x1, y1, and after that put the chosen terms in the general equation of the point-slope form. And then rearrange the linear equation into the standard form.
Example 1
Calculate the equation of the line by using point-slope form have points (3, -6) and the slope m 21?
Solution
Step 1: Identify the fixed points x1, y1 and the slope m.
x1 = 3
y1 = -6
m = 21
Step 2: Write the general formula of point-slope form.
y – y1 = m (x – x1)
Step 3: Put the point and slope in the above formula.
y – y1 = m (x – x1)
y – (-6) = 21 (x – 3)
y + 6 = 21x – 63
y + 6 – 21x + 63 = 0
y – 21x + 69 = 0
Step 4: Convert the above equation into standard form.
-(y – 21x + 69) = -(0)
-y + 21x – 69 = 0
21x – y – 69 = 0
Example 2
Calculate the equation of the line by using point-slope form have points (-13, 3) and the slope m 11?
Solution
Step 1: Identify the fixed points x1, y1 and the slope m.
x1 = -13
y1 = 3
m = 11
Step 2: Write the general formula of point-slope form.
y – y1 = m (x – x1)
Step 3: Put the point and slope in the above formula.
y – y1 = m (x – x1)
y – (3) = 11 (x – (-13))
y – (3) = 11 (x + 13)
y – 3 = 11x – 143
y – 3 – 11x + 143
y – 11x + 140 = 0
Step 4: Convert the above equation into standard form.
-(y – 11x + 140) = -(0)
-y + 11x – 140 = 0
11x – y – 140 = 0
Other methods to find the equation of the line
Some other methods are widely used to find the equation of the line. These methods are very famous and very useful for the calculation of the equation of the line. The other methods are.
- Slope intercept form
- Two-point form
- Intercept from
Slope Intercept Form
This is another type to find the equation of the line. This type use slope and the intercept form of the line to find the equation of the line. The equation of this type is written as.
y = mx + b
In the above equation, x and y are the random variables, m is the slope of the line, and b is the y-intercept of the given line.
Example
Calculate the equation of the line by using slope-intercept form having points (3, 9), (4, -6).
Solution
Step 1: Identify the values that are given in the problem.
x1 = 3
x2 = 4
y1 = 9
y2 = -6
Step 2: Write the general formula to calculate the equation of the line of the slope-intercept form.
y = mx + b
Step 3: First of all, find the slope by using given points.
m = y2 – y1 / x2 – x1
m = -6 – 9 / 4 – 3
m = -15/1
m = -15
Step 4: Now calculate y-intercept form by using any pair of the points that are given.
y = mx + b
y = -15x + b
put (3, 9) in the above equation.
9 = -15(3) + b
9 = -45 + b
b = 45 + 9
b = 54
Step 5: Put the value of slope and y-intercept form in the general formula of the slope-intercept form.
y = MX + b
y = -15x + 54
Hence, this is the required equation of the line.
Two-point Form
This type of equation to find the equation of the line using an equation having 2 points and general fixed points.
y – y1 = (y2 – y1 / x2 – x1) (x – x1)
Intercept Form
This type of equation to find the equation of line used an equation of the form.
x/a + y/b = 1
Summary
The equation of the line is generally calculated by various methods such as point-slope form, slope-intercept form, two-point form, and intercept form. These techniques are used to calculate the equation of the line. For the calculation of the equation of the line, the knowledge of slope and points is very essential.
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