The linear equation of the line is a basic concept of mathematics that is widely used in geometry. Geometry is a fundamental branch of mathematics that deals with the study of properties of lines, shapes, and points.
The straight-line equation is a fundamental concept in mathematics and is used in many different areas, such as physics, engineering, and economics. In this article, we will take a look at the definition of the equation of a line, the different forms of the equation, and how to find the equation of a line.
What is the Equation of a Line?
In geometry, the linear equation of a line is that equation that explains the relationship among the coordinates of the x-axis and y-axis of each and every point on the line. In simple words, the equation of the line tells us how much the coordinate points of the y-axis change as we move along the line in the negative or positive coordinate points of the x-axis.
Forms of Equation of the Line
The equation of a straight line in a 2-dimensional Cartesian coordinate system can be represented in different forms. Such as:
- Slope intercept form
- Point slope form
- Standard form of a line
Let’s explore the forms of the straight-line equation.
Slope intercept form
Slope intercept form is the most commonly used form to write the linear equation of the line in the form of:
Y = m * X + c
Where
- “m” represents the slope of the line
- X & Y represents fixed points of the line
- “c” represents the y-intercept of the line
This is the general equation of slope intercept form in which the value of the slope and y-intercept should be placed to make the equation of the line. If the slope and y-intercept are not given, then you should evaluate them first.
For doing this knowledge about these measurements is very essential.
What is slope?
A slope is a measure of steepness (aka sharpness) of a line. It tells us how much the coordinate points of the y-axis change as we move along the line in the negative or positive coordinate points of the x-axis. The value of the slope of the line could be negative, zero, positive, or undefined.
- Positive slope: When the value of slope is positive it means the line will move upward from the right.
- Negative slope: When the value of slope is negative it means the line will move downward from the right.
- Zero slopes: When the value of the slope is zero it means the vertical line (y-axis) is zero and the line belongs to the horizontal line.
- Undefined Slope: When the value of the slope is undefined it means the horizontal line (x-axis) is zero and the line belongs to the vertical line.
The slope “m” of the line could be evaluated with the help of the formula as the change in the values of the y-axis over the change in the values of the x-axis.
Slope = m = rise / run
m = [y2 – y1] / [x2 – x1]
Where
- (x1, y1) and (x2, y2) are two points on the line.
You can use a slope calculator to evaluate the slope of the line through given points of the line according to the formula.
What is the Y-intercept?
In geometry, the y-intercept of the line is that point that cuts the y-axis of the line. At the point of the y-intercept, the x-coordinate values will be zero. It can be evaluated easily by putting zero as the value of “x” in the equation of the slope-intercept form.
How to find the equation of a line using slope intercept form?
There are different ways to find the equation of the line using slope-intercept form.
For Two Coordinate Points of Line
Example:
Find the straight line equation through slope intercept form with the help of the given coordinate points of the x & y axis of the line.
(x1, y1) = (3, 4) & (x2, y2) = (45, 67)
Solution
Step 1: Write the given coordinate points of the line.
- x1 = 3
- x2 = 45
- y1 = 4
- y2 = 67
Step 2: As the slope of the line is not given, evaluate it first with the help of the above points of the line using the slope formula.
m = [y2 – y1] / [x2 – x1]
Trending Online Courses in India 2023
| Online BA | Online MA |
| Online BBA | Online MBA |
| Online BCA | Online MCA |
| Online BSC | Online MCOM |
| Online BCOM | Online Diploma |
Put the given values
Slope = m = [67 – 4] / [45 – 3]
Slope = m = [63] / [42]
Slope = m = [21] / [14]
Slope = m = 3/2
Slope = m = 1.5
Hence, the slope of the line is positive means that the line goes down as we move to the right.
Step 3: Now find the y-intercept of the line by placing the value of coordinate points and the slope of the line in the equation of slope-intercept form.
y = mx + c
4 = 3/2(3) + c
4 = 9/2 + c
4 – 9/2 = c
4 – 4.5 = c
-0.5 = c
Step 4: Now place the value of the slope of the line “m” & y-intercept of the line “c” in the equation of the slope-intercept form to find the line equation.
y = mx + b
y = 1.5x + (-0.5)
y = 1.5x – 0.5
For 1 Point & Slope
Example
Find the straight line equation through slope intercept form with the help of the given coordinate point and slope of the line.
(x1, y1) = (2, 8) & slope = -15
Solution
Step 1: Take the given data of the line.
- x1 = 2
- y1 = 8
- m = -15
Step 2: Now find the y-intercept of the line by placing the value of coordinate points and the slope of the line in the equation of slope-intercept form.
y = mx + c
8 = -15(2) + c
8 = -30 + c
8 + 30 = c
38 = c
Step 3: Now place the value of the slope of the line “m” & y-intercept of the line “c” in the equation of the slope-intercept form to find the line equation.
y = mx + b
y = -15x + (38)
y = -15x + 38
For Slope and Y-intercept
Example 1:
Find the straight line equation through slope intercept form with the help of the given y-intercept of the line and slope of the line.
b = -2.4 & slope = 9
Solution
Step 1: Take the given data of the line.
b = -2.4
m = 9
Step 2: Now place the value of the slope of the line “m” & y-intercept of the line “c” in the equation of the slope-intercept form to find the line equation.
y = mx + b
y = 9x + (-2.4)
y = 9x – 2.4
You can also use online tools like a slope-intercept form calculator by AllMath, to determine the equation of a line, similar to the ways mentioned above.
Point Slope Form
Point slope form is the commonly used form to write the straight line equation in the form of:
y – y1 = m (x – x1)
In this form,
- (x1, y1) is a point on the line.
- “m” represents the slope of the line
- x & y represents fixed points of the line
This is the general equation of point slope form in which one point on the line and the slope of the line should be placed to find the equation of a line. If the slope is not given, then you should evaluate it first.
How to find the equation of a line using point slope form?
There are different ways to find the equation of the line using the point-slope form.
For two points
Example 1: Positive points
Find the straight line equation through point slope form with the help of the given coordinate points of the x & y axis of the line.
(x1, y1) = (2, 10) & (x2, y2) = (60, 70)
Solution
Step 1: Write the given coordinate points of the line.
- x1 = 2
- x2 = 60
- y1 = 10
- y2 = 70
Step 2: As the slope of the line is not given, evaluate it first with the help of the above points of the line using the slope formula.
m = [y2 – y1] / [x2 – x1]
Put the given values
Slope = m = [70 – 10] / [60 – 2]
Slope = m = [60] / [58]
Slope = m = [30] / [29]
Slope = m = 1.03
Hence, the slope of the line is positive means that the line goes down as we move to the right.
Step 3: Now take the general expression of the point-slope form.
y – y1 = m (x – x1)
Step 4: Now find the straight line equation by placing the value of coordinate points and the slope of the line in the equation of the point-slope form.
y – y1 = m (x – x1)
y – (10) = 1.03 * (x – 2)
y – 10 = 1.03x – 1.03 * 2
y – 10 = 1.03x – 2.06
y – 10 – 1.03x + 2.06 = 0
y – 1.03x + 7.94 = 0
1.03x – y – 7.94 = 0
Example 2: Negative points
Find the straight line equation through point slope form with the help of the given coordinate points of the x & y axis of the line.
(x1, y1) = (-5, -9) & (x2, y2) = (20, 41)
Solution
Step 1: Write the given coordinate points of the line.
- x1 = -5
- x2 = 20
- y1 = -9
- y2 = 41
Step 2: As the slope of the line is not given, evaluate it first with the help of the above points of the line using the slope formula.
m = [y2 – y1] / [x2 – x1]
Put the given values
Slope = m = [41 – (-9)] / [20 – (-5)]
Slope = m = [41 + 9] / [20 + 5]
Slope = m = [50] / [25]
Slope = m = [10] / [5]
Slope = m = 2
Hence, the slope of the line is positive means that the line goes down as we move to the right.
Step 3: Now take the general expression of the point-slope form.
y – y1 = m (x – x1)
Step 4: Now find the straight line equation by placing the value of coordinate points and the slope of the line in the equation of the point-slope form.
y – y1 = m (x – x1)
y – (-9) = 2 * (x – (-5))
y + 9 = 2 * (x + 5)
y + 9 = 2x + 2 * 5
y + 9 = 2x + 10
y + 9 – 2x – 10 = 0
y – 2x – 1 = 0
2x – y + 1 = 0
For 1 Point and Slope
Example
Find the straight line equation through point slope form with the help of the one point & slope method if the slope of a line is 8 and the coordinate point is (8, 2).
Solution
Step 1: Take the given values of the slope and coordinate points of a line.
- Slope = m = 8
- x1 = 8
- y1 = 2
Step 2: Now take the general expression of the point-slope form.
y – y1 = m (x – x1)
Step 3: Now find the straight line equation by placing the value of coordinate points and the slope of the line in the equation of the point-slope form.
y – y1 = m (x – x1)
y – 2 = 8 * (x – 8)
y – 2 = 8 * x – 8 * 8
y – 2 = 8x – 64
y – 2 – 8x + 64 = 0
y – 8x + 62 = 0
8x – y – 62 = 0
Applications of the Equation of a Line
The real-world applications of the equation of the line are as follows:
- The equation of the straight line is used to model the connection among two variables i.e., the number of units sold and the price of the ingredient.
- The straight line equation is essential for solving the problems of motions i.e., finding the space covered by a bus over a period of time.
- Line equation is used to explain the properties of the line in geometry i.e., steepness of the line, angle with axes, and the points of intersections.
- It is also used for the modeling of the linear motion i.e., the connection between displacement and the time in constant velocity circumstances.
- The equation of the straight line can also be used for financial modeling and risk management to expect trends and analyze connections among variables.
Wrap Up
The equation of a line is a mathematical equation that describes the relationship between the x- and y-coordinates of all the points on the line. There are many different ways to write the equation of a line, but the most common form is the slope-intercept form and point-slope form.
The slope of a line tells us how steep the line is, and the y-intercept tells us the point at which the line crosses the y-axis. The equation of a line has many applications in the real world, such as modeling the relationship between two variables and solving problems involving motion.