Online Slope Intercept form Calculator
How to use the slope intercept form calculator
To use the point-slope form calculator, follow the procedure given below.
- Enter the coordinates of any two points of the given line in the blanks provided here.
- Click on the ‘solve’ button.
- The slope intercept form calculator will provide you with the following data:
3.1 Slope
3.2 y-intercept
3.3 Equation of the line
3.4 Graphical representation of the line
What is a Line?
A line is a figure made up of a set of points that extends to both sides infinitely. It is also defined as a one-dimensional figure that has no width.
What is a slope intercept form?
The slope of a line shows the steepness of the line, and it is represented using the letter “m”. The slope tells us how the value of “y” changes for a change in “x”. The y-intercept is the y-coordinate of the point where the line intersects the y-axis. The y-intercept is represented by “b”. The equation of a line in the slope intercept form is y=mx+b
Derivation of the slope intercept form using two given points:
Let’s assume that the coordinates of two points on a line are \((x_1,y_1)\) and \((x_2,y_2)\).
Suppose the equation of the given line is y=mx+b.
Since both of the points pass through the line,
- \(y_1 = mx_1+b\)
- \(y_2 = mx_2+b\)
Then, after subtracting the first equation from the second equation,
\(y_2-y_1 = m(x_2-x_1)\)
m = \(\frac{y_2-y_1}{x_2-x_1}\)
Substitute the value of “m” into the first or second equation to find the equation for the y-intercept. Taking the first equation we get,
\(y_1 = mx_1+b\)
b = \(y_1-mx_1\)
Taking the second equation we get
b = \(y_2-mx_2\)
Solved examples relevant to applying the slope intercept formula
Example 1 :
Find the equation of the line passing through the points (6,9) and (7,3).
Solution:
The coordinates of the points in the question are:
\(x_1 = 6,\)
\(x_2 = 7,\)
\(y_1 = 9,\)
\(y_2 = 3\)
The formula to find the value of the slope, “m” is:
m = \(\frac{y_2-y_1}{x_2-x_1}\)
m = \(\frac{3-9}{7-6}\)
m = \(\frac{-6}{1}\)
= -6
The formula to find the value of “b”.
b = \(y_1-mx_1\)
b = 9- \((-6)\times6\)
= 9 – [-36]
= 45
So, the equation of the line passing through the two points are:
y = mx+b
y = -6x+45
Example 2:
The equation that represents the cost of taking a taxi is given as
y = 3x+4
Here, “x” represents the distance traveled in miles, and “y” represents the cost of traveling the given distance in dollars. Explain the meaning of slope in this context. What is the difference in cost when traveling 100 miles and 150 miles?
Solution:
We will first figure out the cost incurred for the distance traveled.
When x = 100,
y = 3x+4
= 3 (100) + 4 = $304
When x = 150,
= 3 (150) + 4 = $454
In this context, the slope represents the increase in the price of the fare as the distance increases. From the given equation, as per the slope intercept formula, the value of m=3. In this context, it means that $3 is the charge for every mile that is covered.
One of the many different forms of the equation of a line is the general form and the standard form. The standard form of the equation of a line is written in the form of, ax+by = c. While the general form of the equation of a line is in the form of ax+by+c = 0. We can calculate the slope, the x-intercept and the y-intercept from these forms of equations of lines.
The standard form of the equation of a line ax+by = c can be written in the slope intercept form as follows,
ax+by = c
by = -ax+c
y = \(-\frac{a}{b}\times x+\frac{c}{b}\)
Compare the above equation with the slope intercept form, y = mx +d
Slope, m = \(\frac{-a}{b}\) and y-intercept, d = \(\frac{c}{b}\)
The standard form of the equation of a line ax+by=c can be written in the slope intercept form as follows,
ax+by+c=0
by = -ax-c
y = \(-\frac{a}{b}\times x-\frac{c}{b}\)
Compare the above equation with the slope intercept form, y = mx +d
Slope, m = \(\frac{-a}{b}\) and y-intercept, d = \(\frac{-c}{b}\)
The slope of any line is calculated by, m = \(\frac{y_2-y_1}{x_2-x_1}\)
Horizontal lines have zero slope because they do not rise vertically but run flat, that is, horizontal lines have the same y-coordinates throughout, which is,(y₂-y₁ = 0).The equation of horizontal lines are y = 0x + b, which is y = b.
In comparison, vertical lines have the same x coordinates throughout, and make the denominator in the slope formula zero,(x₂-x₁ = 0). This results in an undefined slope of vertical lines as a value divided by zero is undefined.