Exploring the Equation of a Line: A Fundamental Concept in Mathematics
In mathematics, the equation of a line is a fundamental topic that is frequently used for analyzing and interpreting linear relationships, making predictions, & solving problems in different disciplines.
There are various ways of representing the equation of the line. in this post, we will learn the basics of the equation of a line, along with its types and examples.
What is an equation of a line?
In mathematical representation, the relation between the coordinate points of x and the coordinate points of y on a line is said to be the equation of a line. In algebraic form, the equation of a line is a way to express the position of a line, the steepness of the line, and some other characteristics.
The straight-line equation can be written in different forms such as:
Slope intercept form
Point slope form
Standard form
These are said to be the methods to write the equation of the line. Let us explore the methods of the equation of a line briefly with the help of solved examples.
What is Slope intercept form?
The slope-intercept form of a line can be written as
y = m*x + c
Where
the symbol “m” represents the slope of the line
“c” represents the y-intercept of the line.
You must have sound knowledge about the slope and y-intercept of the line for dealing with the slope-intercept form equation. Let us describe these concepts briefly.
The slope of the line is the measure of the steepness of the line. It is the change in the value of the y-axis over the change in the values of the x-axis. The formula to find the inclination of the line is:
Slope = m = y2 – y1 / x2 – x1
While the y-intercept of the line is that point that intersects the line at the y-axis. It is the starting value of y at x is zero.
Example 1: For two coordinate points
Find the straight line equation if the coordinate points of x & y are (x1, y1) = (15, 8) & (x2, y2) = (35, 38) with the help of the slope-intercept equation.
Solution
Step 1: Write the given point of the x-axis and y-axis of the line.
x1 = 15, x2 = 35, y1 = 8, y2 = 38
Step 2: Now find the slope of the line by using its formula and putting the points of the x-axis and y-axis of the line.
Step 3: Now find the y-intercept of the line by placing the slope and points of the line.
y = mx + b
8 = 3/2(15) + b
8 = 45/2 + b
8 – 45/2 = b
8 – 22.5 = b
b = -14.5
Step 4: Now place the results of the slope and y-intercept of the line to find the straight line equation using slope-intercept form.
y = mx + b
y = 1.5x + (-4.5)
y = 1.5x – 4.5
Example 2: For 1 point & slope
Find the straight line equation if the coordinate points of x & y and the slope of the line are (x1, y1) = (14, 13) and -15 respectively with the help of the slope-intercept equation.
Solution
Step 1: Write the given point of the x-axis and y-axis of the line and the slope of the line.
x1 = 14
y1 = 13
m = -15
Step 2: Now find the y-intercept of the line by placing the slope and points of the line.
y = mx + b
13 = -15(14) + b
13 = -210 + b
13 + 210 = b
223 = b
Step 3: Now place the results of the slope and y-intercept of the line to find the straight line equation using slope-intercept form.
y = mx + b
y = -15x + 223
The problems of finding the equation of a line using the slope-intercept form can also be solved using a y=mx+b calculator, which helps to avoid lengthy calculations.
What is point-slope form?
The point-slope form of a line can be written as
y – y1 = m * (x – x1)
Where,
“m” represents the slope of the line
x & y are the fixed points of the line
x1 & y1 are the coordinate points of the line.
This form of the equation of the line is totally dependent on the slope and the point of the line. It describes the relationship between the slope and a specific point on the line.
Example 1: For two points
Find the straight line equation if the coordinate points of the line are (x1, y1) = (12, 7) & (x2, y2) = (21, 28) with the help of point slope form.
Solution
Step 1: Write the given point of the x-axis and y-axis of the line.
x1 = 12, x2 = 21, y1 = 7, y2 = 28
Step 2: Now find the slope “m” of the line by taking the general formula of finding the slope of the line or taking assistance from a slope finder.
Using the slope finder the result will be:
Step 3: Now take the general expression of the point-slope form and put one point and slope “m” of the line to it to evaluate the straight-line equation.
Example 2: For 1 point and slope
Find the straight line equation if the coordinate points and slope of the line are (x1, y1) = (5, 2) and 3 respectively with the help of point slope form.
Solution
Step 1: Write the given point of the x-axis and y-axis of the line and the slope of the line.
Slope of a line = m = 3
x1 = 5
y1 = 2
Step 2: Now write the general form of the point-slope form.
y – y1 = m (x – x1)
Step 3: Now take the general expression of the point-slope form and put one point and slope “m” of the line to it to evaluate the straight-line equation.
What is the standard form?
The standard form of a line can be written as
Ax + By = C
Where
x & y are the fixed points of the line
A, B, & C are constants of the line.
The standard form of the line presents the straight line equation with the coefficients and these forms can be represented as a vertical line when the constant B is zero.
Conclusion
The equation of the line is very essential in mathematics for dealing with the various aspects of daily life and problems. You can grab all the basics of the equation of the line from this post as we have discussed all the basics with the help of solved examples.