How To Find Slope for Two Points

Understanding whether the given lines are perpendicular or parallel without the use of any geometrical tool can be done by the measurement of the slope. To understand the concept of finding the slope from two points worksheet and finding the slope worksheet answers in a deeper and more detailed manner at first, it is important to understand the meaning of slope.

Meaning of slope: With respect to mathematics, a line with the change in the coordinate ‘y’, with respect to the change in the coordinate ‘x’ is known as a slope. Δy is used to identify the net change in the y-coordinate and Δx is used to identify the net change in the x-coordinate. Thus, the equation for the coordinate of y with respect to the coordinate of x can be denoted by,

m= change in y coordinate/ change in on x coordinate. The equation in the simplest form can be written as, Δy/ Δx.

In the equation, ‘m’ represents the slope of the line. The slope of the line can even be shown in the following way,

tan θ = Δy/ Δx

Here, tan θ is the slope of the line. 

We should understand that usually, the slope of the line gives the measure of its direction and steepness. By finding the difference between the coordinates of the points, the determining the slope of a straight line between the two points say, (x1,y1) and (x2,y2). Generally, the slope is determined by ‘m’.

The formula to find the slope from two points worksheet.

If the two points of the slope on the straight line are denoted by P (x1,y1) and Q (x2,y2) then the formula for slope can be written as,

Slope m = change in coordinates of y / change in coordinates of x.

In simple words, (y2 – y1) (x2 – x1)

Based on the above formula it is convenient to calculate the slope of a line between the two points. In other words, the slope of the line between the two points can also be mentioned as over the run, the rise of the line from one point to another. Thus slope can be written as,

m= Rise/Run

The equation for the slope of the line.

The slope of the line equation and the points are also known as a point-slope form of the equation of the st. the line is represented as,

y-y1 =m(x-x1) 

The equation of the line for a slope-intercept form of the equation is represented as, 

y= mx+b

Here, the variable ‘b’ is the intercept of y.

Finding the slope of a line on the graph.

If the slope of the line is m = tan θ and if the two points A (x1,y1) and B (x2,y2) are on the line then the variable x2 is not equal to x1 and thus the slope of the line is denoted as,

m = tan θ = y2-y1 / x2-x1 

In the above equation, θ is the angle that the line AB makes the direction, positive to the x-axis. Here, θ lies between 0 degrees and 180 degrees. 

It must be considered that when the line is parallel to the y-axis only then θ becomes 90 degrees. Here the variable x1 becomes unequal to the variable x2 and thus the slope of the line becomes undefined. 

There are three different ways of finding the slope of the straight line. They are parallel, collinearity, and perpendicular. The conditions of the straight line for parallel and collinearity are been mentioned below.

The slope for the Parallel Line.

If we consider ‘l1’ and ‘l2’ as the two parallel lines with α and β as the inclinations respectively then for the parallel lines mentioned their inclinations must also be equal. Thus, it can be denoted as α = β. Thus, the conditions for two lines with inclinations alpha (α) and beta (β) is that both should be equal to each other, i.e, α = β. 

Thus, on the cartesian plane, if the slope of the two lines is equal then both the line are parallel to each other. 

Then, if two lines are parallel then, m1 = m2. 

The slope for the collinearity Line.

The slope of both the lines must be equal for the lines to be collinear. Also, there should be at least one common point through which they should pass by. 

Positive and Negative Slope.

If the line goes up as we move along then it is known for the value of the slope of the line to be positive. Thus, we can say that the rise over run is positive.

As we move along the x-axis, if the line goes down in the graph then the value of the slope is negative. 

Examples for the slope of the line:

Find the slope of the line between the points P = (0,-1) and Q = (4,1)

The solution for the above stated problem has been prescribed in the below mentioned steps,

Given: P = (0, -1) and Q = (4, 1)

m = (y2-y1) (x2-x1) {According to the formula}

Therefore, m = (1-(-1)) / 4-0 = 2/4 = 0.5

Conclusion:

Thus, we learned about the two methods of finding the slope for the straight line and also understood the positive slope and negative slope along with the difference between them. And thus with the help of the examples mentioned above, we were able to derive the solution for finding the slope from two points worksheet and finding the slope worksheet answers.