Point Slope Form of a Line Calculator

Point slope form calculator uses coordinates of a point `A(x_A,y_A) `and slope m in the two- dimensional Cartesian coordinate plane and find the equation of a line that passes through A. This tool allows us to find the equation of a line in the general form Ax + By + C = 0. It’s an online Geometry tool requires one point in the two-dimensional Cartesian coordinate plane and coefficient m.
It is necessary to follow the next steps:

1. Enter coordinates `(x_A,y_A)` of two point A and the coefficient for the slope m in the box. These values must be real numbers or parameters.
2. Press the "GENERATE WORK" button to make the computation.
3. Point slope form calculator will give the equation of line in the general form. This line passes through the point A and has the slope m.

Input: An ordered pair of real numbers or variables as coordinates of a point and a real number or variable as coefficient of a slope.
Output: An equation of a line in the general form, Ax + By + C = 0.

Slope Formula:

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

In other words, the formula for the slope can be written as

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:

1. Positive slope `m > 0`, if a line `y = mx + b` is increasing, i.e. if it goes up from left to right
2. Negative slope `m < 0`, if a line `y = mx + b` is decreasing, i.e. if it goes down from left to right
3. Zero slope, `m = 0`, if a line `y = mx + b` is horizonal. In this case, the equation of the line is `y = b`
4. 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.

How to Find Point Slope Form of Line?

If we plug in the coordinates `x_A` and `y_A` and the given value of slope m, into the equation `y − y_A = m(x − x_A)`, we will obtain the equation of the line that passes through the point `A(x_A,y_A)` and has the slope m. In many cases, we can find the equation of the line by hand, especially for integers. But, if the input values are big real number or number with many decimals, then we should use the point slope form calculator to get an accurate result.

The point slope work with steps shows the complete step-by-step calculation for finding the general equation of line through point A at coordinates (5,4) with the slope m = 8. For any other combinations of point and slope, just supply the coordinates of point and slope coefficient and click on the "GENERATE WORK" button. The grade school students may use this point Slope calculator to generate the work, verify the results or do their homework problems efficiently.

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.

Slope Practice Problems

Practice Problem 1:
A tree grows at steady rate of 15 inches per day and achieves its full height of 500 inches in 30 days. Write the general equation of this linear model.

Practice Problem 2:
The slope of line is −5 and line passes through point (4,0). Find the general equation of the line.

The point slope form calculator, formula, example calculation (work with steps) and practice problems would be very useful for grade school students (K-12 education) to learn what are different equations of a line in geometry, how to find the general equation of a line. They will be able to solve real-world problems using linear models in point slope form.