Define What is the Slope of Line?
The slope of a line in the two-dimensional Cartesian coordinate plane is usually represented by the letter m, and it is sometimes called the rate of change between two points. This is because it is the change in the y-coordinates divided by the corresponding change in the x-coordinates between two distinct points on the line. If we have coordinates of two points `A(x_A,y_A)` and `B(x_B,y_B)` in the two-dimensional Cartesian coordinate plane, then
the slope m of the line through `A(x_A,y_A)` and `B(x_B,y_B)` is fully determined by the following formula
`m=\frac{y_B-y_A}{x_B-x_A}`
In other words, the formula for the slope can be written as
$$m=\frac{\Delta y}{\Delta x}=\frac{{\rm vertical \; change}}{{\rm horizontal \; change}}=\frac{{\rm rise}}{{\rm run}}$$
As we know, the Greek letter `∆`, means difference or change. The slope m of a line `y = mx + b` can be defined also as the rise divided by the run. Rise means how high or low we have to move to arrive from the point on the left to the point on the right, so we change the value of `y`. Therefore, the rise is the change in `y`, `∆y`. Run means how far left or right we have to move to arrive from the point on the left to the point on the right, so we change the value of `x`. The run is the change in `x`, `∆x`.
The slope m of a line `y = mx + b` describes its steepness. For instance, a greater slope value indicates a steeper incline. There are four different types of slope:
- Positive slope `m > 0`, if a line `y = mx + b` is increasing, i.e. if it goes up from left to right
- Negative slope `m < 0`, if a line `y = mx + b` is decreasing, i.e. if it goes down from left to right
- Zero slope, `m = 0`, if a line `y = mx + b` is horizonal. In this case, the equation of the line is `y = b`
- Undefined slope, if a line `y = mx + b` is vertical. This is because division by zero leads to infinities. So, the equation of the line is `x = a`. All vertical lines `x = a` have an infinite or undefined slope.
Real World Problems Using Point Slope of a Line
As we mentioned, the fundamental applications of slope or the rate of change are in geometry, especially in analytic geometry. But, the rate of change is also fundamental to the study of calculus. For non-linear functions, the rate of change varies along the function. The first derivative of the function at a point is the slope of the tangent line to the function at the point. So, the first derivative is the rate of change of the function at the point.
In physics, in definitions of some magnitudes such as displacement, velocity and acceleration, the rate of change play important role. For instance, the rate of change of a function is connected to the average velocity.
The rate of change can be found also in many fields of life, for instance population growth, birth and death rates, etc.
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- Home / Point Slope Form Calculator
Enter the values of X1, Y1, and (m) in the below input boxes and hit the Calculate button to get the equation of a straight line using point slope form calculator.
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Point slope calculatoris an online tool developed to find the equation of a straight line using the slope and point on that line.
What is point-slope?
Point slope form is a general form for linear equations. It highlights the slope of the line and a point on the line. You can find the equation in the below section.
Point slope formula
The point slope equation can be expressed as:
y - y1 = m(x- x1)
Where,
m is the slope, and
x1, y1 are the coordinates of a point.
How to find equation of a line?
To find equation of a straight line without a slope intercept form calculator,follow the below example.
Example:
Find the equation of a line whose known points are (2, 3) and slope of line is 8.
Solution:
Step 1:Identify and write down the values.
x1 = 2
y1 = 3
m = 8
Step 2: Place the values in the point slope formula and solve the equation.
y - y1 = m(x- x1)
y – 3 = 8(x - 2)
y – 3 = 8x – 16
8x – 16 – y + 3 = 0
8x – 13 – y = 0
y = 8x – 13
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