I. Introduction
If math is not your forte, finding the equation of a line can be a daunting task. The many different formulas and methods can leave anyone confused. However, don’t fret because we’ve studied and tested these five foolproof methods to find the equation of a line to help you master the basics of math.
II. 5 Foolproof Methods to Find the Equation of a Line
Before we dive in, let us first familiarize ourselves with the different methods we will cover. The five methods that we will discuss are:
- Slope-intercept form
- Point-slope form
- Two-point form
- Standard form
- Intercept form
A. Method #1: Slope-intercept form
The slope-intercept form is probably the most familiar method in the list. It is a simple equation where you can find the slope and y-intercept given two points.
1. Explanation of the formula
The formula for slope-intercept form is:
y = mx + b
where:
- y is the y-value
- m is the slope
- x is the x-value
- b is the y-intercept
2. Examples of how to use it
Example 1:
Given two points (2,5) and (4,9), find the slope and y-intercept of the line equation.
Step 1: Find the slope
m = (y2 - y1) / (x2 - x1)
m = (9 - 5) / (4 - 2)
m = 2
Step 2: Find the y-intercept
b = y - mx
b = 5 - (2)(2)
b = 1
Therefore, the line equation is:
y = 2x + 1
Example 2:
Given two points (-3,2) and (5,8), find the slope and y-intercept of the line equation.
Step 1: Find the slope
m = (y2 - y1) / (x2 - x1)
m = (8 - 2) / (5 - (-3))
m = 6 / 8
m = 3 / 4
Step 2: Find the y-intercept
b = y - mx
b = 2 - (3/4)(-3)
b = 2 + 9/4
b = 17/4
Therefore, the line equation is:
y = (3/4)x + 17/4
B. Method #2: Point-slope form
The point-slope form is perfect to use if you have one point and the slope of the line you want to find the equation for. It’s a simple formula but can be confusing for those who have never used it before.
1. Explanation of the formula
The formula for point-slope form is:
y - y1 = m(x - x1)
where:
- y is the y-value
- y1 is the y-value of the point
- m is the slope
- x is the x-value
- x1 is the x-value of the point
2. Examples of how to use it
Example 1:
Given the point (3,7) and the slope 2/3, find the equation of the line.
y - y1 = m(x - x1)
y - 7 = (2/3) (x - 3)
y - 7 = (2/3)x - 2
y = (2/3)x + 5
Therefore, the line equation is:
y = (2/3)x + 5
Example 2:
Given the point (-2,-5) and the slope -4, find the equation of the line.
y - y1 = m(x - x1)
y - (-5) = -4(x - (-2))
y + 5 = -4(x + 2)
y = -4x - 13
Therefore, the line equation is:
y = -4x - 13
C. Method #3: Two-point form
The two-point form is a useful formula when you have two points, and you want to find the equation of the line. This method is an excellent solution when you don’t know the slope and y-intercept of the line.
1. Explanation of the formula
The formula for the two-point form is:
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
where:
- y is the y-value
- y1 is the y-value of the first point
- y2 is the y-value of the second point
- x is the x-value
- x1 is the x-value of the first point
- x2 is the x-value of the second point
2. Examples of how to use it
Example 1:
Given two points (1,3) and (4,7), find the equation of the line.
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
(y - 3) / (x - 1) = (7 - 3) / (4 - 1)
(y - 3) / (x - 1) = 4 / 3
y - 3 = (4/3)(x - 1)
y - 3 = (4/3)x - 4/3
y = (4/3)x + 5/3
Therefore, the line equation is:
y = (4/3)x + 5/3
Example 2:
Given two points (-1,5) and (3,-1), find the equation of the line.
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
(y - 5) / (x - (-1)) = (-1 - 5) / (3 - (-1))
(y - 5) / (x + 1) = -6 / 4
(y - 5) / (x + 1) = -3 / 2
y - 5 = (-3/2)(x + 1)
y - 5 = (-3/2)x - 3/2
y = (-3/2)x + 7/2
Therefore, the line equation is:
y = (-3/2)x + 7/2
D. Method #4: Standard form
The standard form is very versatile because the coefficients of x and y are both integers. It’s also easy to graph because of its unique simplicity, but it’s not the simplest to derive.
1. Explanation of the formula
The formula for standard form is:
Ax + By = C
where:
- A and B are integers
- x is the x-value
- B is the y-value
- C is an integer
2. Examples of how to use it
Example 1:
Given the slope 4/5 and y-intercept -8/5, find the line equation in standard form.
We first need to rearrange the equation to make A, B, and C integers:
y = mx + b
y = (4/5)x - 8/5
5y = 4x - 8
4x - 5y = 8
Therefore, the line equation in standard form is:
4x - 5y = 8
Example 2:
Given two points (0,-3) and (2,1), find the line equation in standard form.