In this platform, you will learn about the slope of a line. The slope of a line calculates the steepness and the direction of the line. It will show how the slant line will be and how much the line rises vertically compared to the runs horizontally. In a coordinate plane predict whether the lines are parallel or perpendicular for finding the slope of the lines.
On this page, check out the 10th Grade Math concept of the slope of a line, formula, how to find the slope of a line with two points, types of slope, methods to find the slope, solved example problems, and so on.
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Slope of a Line Meaning & Definition
In mathematics, the slope of a line is defined as the change in the y coordinate with respect to the change in the x coordinate of that line. The net change in the y coordinate is Δy, while the net change in the x coordinate is Δx. So the change in the y coordinate with respect to the change in the x coordinate can be written as
m = change in y / change in x = Δy/Δx.
tan θ = Δy/Δx.
where tan θ, m will be the slope of the line.
It is also defined as the ratio of the rise to the run, or rise divided by the run. The slope of a line measures the tangent of the angle made by the line with the x-axis. In general, to determine a line’s slope, we need to have two different coordinates values on the line. The slope of a line image is shown below,
The slope is constant throughout a straight line. To calculate the slope of a line, the x and the y coordinates of the points lying on the line can be used. To calculate slope of a line the formula is
m = (y2 – y1)/(x2 – x1) = Δy/Δx
Where x1, x2 are the coordinates of the x-axis and y1, y2Â are the coordinates of the y-axis.
Slope Between Two Points
In a straight line, there are two points by using these two points we can calculate the slope of a line. We can apply the coordinates of the two points, in a slope of line formula. The two coordinate points are,
P1 = (x1, y1)
P2 = (x2, y2)
We know that the slope of a line is the change in the y coordinate with respect to the change in the x coordinate of that line. So, substitute the values of Δy and Δx in the equation of slope, then we get
Δy = y2 – y1
Δx = x2 – x1
Now, these values in a ratio, we get
The Slope m = tan θ = (y2 – y1)/(x2 – x1)
where the slope of a line is m.
θ is the positive x-axis line angle.
Slope of a Line Formula
Using the equation of the line we will calculate the slope of a line. The slope of a line general formula is given as,
y = mx + b
Where m is the slope of a line, m = tan θ = Δy/Δx.
θ, be the angle made by the line with the positive x-axis.
Δy is the net change in the y-axis.
Δx is the net change in the x-axis.
Example: What is the equation of a line whose slope is 1, that passes through the point (-2, -6)?
Solution: Given that the slope is 1. So the value of m will be 1.
We know the general equation of a line formula, y = mx + b.
Now substitute the value of m is 1, then we get
y = x + b.
We have ‘a’ the value of one point on the line. So, we put the value of the point (-2, -6) in the equation y = x + b, and we get,
b = -4
Substituting the values of m and b in the general equation, we get the final equation y = x – 4.
Therefore, the equation of a line is y = x – 4.
How do you find the Slope of a Line?
Using the different methods we will find the slope of the line. The first method is the equation method, using this method to find the value of the slope of a line. The equation is,
m = (y2 – y1)/(x2 – x1)
where m is the slope of the line.
In other words, the change in x is run, and therefore the change in y is rise or fall. Thus, we defined a slope of a line as, m = rise/run.
How to Find Slope of a Line on Graph?
You are finding the slope of a line from the graph, one method is directly applied which is the formula method to the coordinates of two points lying on the line. If the values of the two coordinates points are not given. At this time, we have to use another method to find the slope of the line. So in this method, we have to find the tangent of the angle which is made by a line on the x-axis. The below figure shows the way how we find the slope of a line,
It has only one value. So, for finding the slopes Method1 and Method 2 will be equal. In addition, we are given the equation of a straight line. The general equation of a line is,
y = mx + b
The value of the slope line is given as m.
The following are the steps to find the slope of a line such that the coordinates of two points lying on the line are (x1, y1), (x2, y2).
Step 1: Note the coordinates of the two points lying on the line, (x2, y2), (x1, y1)
Step 2: Apply the slope of line formula, m = (y2 – y1)/(x2 – x1).
Step 3: Finally, we get the value of the given slope line.
Types of Slope
Depending upon the relationship between the two variables x and y there are different types of slopes and then the value of the gradient or slope of the line is obtained. There are 4 different types of slopes namely,
-
- Positive slope
- Negative slope
- Zero slope
- Undefined Slope
Positive Slope: A positive slope indicates that in a coordinate plane while moving from left to right the line rises, it also signifies that when x increases, the y also increases.
Negative Slope: In a negative slope, the coordinate plane is moving from left to right then the line falls, which also signifies that when x increases, then y decreases.
Zero Slope: The line in a zero slope the rise will be zero. So applying the rise over run formula we get the slope of the line as zero.
Undefined Slope: In an undefined slope, the line value of the run is zero. So, the slope of a vertical line will be undefined.
The Slope of the Horizontal Line
A horizontal line is a straight line that is parallel to the x-axis or in a coordinate plane, it will be drawn from left to right or right to left. So, the net change in the y-coordinates of the horizontal line will be zero. Then the slope of a horizontal line will be,
Slope of a horizontal line, m = Δy/Δx = zero.
The Slope of the Vertical Line
The vertical line is a straight line that is parallel to the y-axis or it is a line from top to bottom or bottom to top in a coordinate plane. Thus, the net change in the x-coordinates of the vertical line is zero. So, the slope of a vertical line is,
The Slope of a vertical line is m = Δy/Δx = undefined.
Finding Slope of the Perpendicular Lines
A set of perpendicular lines always has an angle of 90º angle between them. Suppose, if we have two perpendicular lines L1 and L2 in the coordinate plane, then the lines are inclined with the x-axis at angles θ1 and θ2 respectively. The given angles will follow the external angle theorem as, θ2 = θ1 + 90º.
Therefore, the slopes can be given as,
m1 = tan θ1
m2 = tan (θ1 + 90º) = – cot θ1
m1 × m2 = -1
So, the two perpendicular lines’ slope product is equal to -1.
Slope of Parallel Lines
A set of parallel lines always has an equal angle of inclination. Suppose we have two parallel lines L1 and L2 in a coordinate plane, then they are inclined at angles θ1 and θ2 respectively with the x-axis. So that θ2 = θ1
Then, the slopes will be
m1= m2
So, the slopes of the two parallel lines are equal.
Read More:
- Drawing Graph of y = mx + c Using Slope and y-intercept
- Problems on Plotting Points in the x-y Plane
- Conditions of Collinearity of Three Points
Slope of a Line Examples
Example 1:
The equation of a line is 2y = 6x + 7, find its slope value?
Solution:
Given that, the equation is 2y = 6x+7.
Now, we need to find out the slope of the equation.
We know that the formula of the slope is, y = mx + b
Now, try to bring the equation to this form. The coefficient of y = 1, then we get,
y = 6x/2 + 7/2 = 3x+3.5
So, the coefficient of x is found to be 3.
Hence, our slope will be the same as the coefficient of x.
Therefore, the slope of an equation line is 4.
Example 2:
If the equation of a line is given as x = 7. Find the value of the given slope of the line?
Solution:
As given in the question, the equation is x = 7.
Now, we will find the value of the slope of the given line.
Observe that y is missing from the given equation. So, we assume that the coefficient of y is 0.
Then we get, (0)y = x – 7
Now, we make the coefficient of y as 1. Let us divide both sides by Zero.
We know that if anyone is divided by zero, then the value can not be determined.
So, in this case, the coefficient of x will be divided by zero which gives us our slope. In such cases, the answer will not be defined. So we can safely say that our slope is not defined in such cases.
The Slope is not defined.
Example 3:
If the rise is 20 units, while the run is just 10 units, What is the slope of the line?
Solution:
In the given question, the run is 10 units and the rise is 20 units.
We all know that the slope of a line will be
m = Rise/Run
Now, substituting the values, we will get
m = Rise/Run = 20/10 = 2
Thus, the slope of a line is 2.
Example 4:
Find the slope of a line that is parallel to the x-axis and intersects the y – axis at y = 2.
Solution:
Given that, the y-axis is y=2.
We all know that the slope of any line is the tangent of its angle made with the x-axis.
So, if the given line is parallel to the x-axis, then the angle will be 0º. So, the tan 0º is 0.
Then the value of the slope is,
m = tan 0 = 0
Thus, the value of the slope will be Zero.
Example 5:
Determine the value of b, if the slope of a line passing through the points (b, 7) and (8, -5) is 6.
Solution:
Given that,
The points are (x1, y1)= (b,7) and (x2 , y2) = (8,-5)
The slope is m= 6.
Now, we will find the value of b.
We know that Slope (m) = (y2 – y1)/(x2 – x1)
Substitute the value in the above formula.
Then the value is 6 = (-5-7)/(8-b)
6 = (-12)/(8-b)
-2= (8-b)
-2-8 = -b
b = 10
Therefore, the value of the b is 10.
FAQ’s on Slope of a Line
1. What is the Slope of a Line?
The slope of a line, also known as the gradient is defined as the value of the steepness or the direction of a line in a coordinate plane. The slope will be calculated by using different methods, given the equation of a line or the coordinates of points lying on the straight line.
2. What is the Slope of a Line formula?
We can calculate the slope of a line directly using the slope of a line formula given the coordinates of the two points lying on the line. The formula is given as,
The slope is m = tan θ =(y2 – y1)/(x2 – x1)
3. How Does Slope Looks Like?
The slope is nothing but the measure of the tangent of the angle made with the x-axis. So, it is a measure of an angle.
4. How do you find the slope of a line?
We need to find the ratio of the difference between the y-coordinates and x-coordinates of the two points, that form the line. The resulted value is the slope of the line. It shows the rise of the line along the y-axis over the run along the x-axis.
5. How do you Show that Three Points are Collinear by Slope?
Using the slope formula we will prove the collinearity of three points. So the slope of lines AB and BC should be equal for the three given points to be collinear.