The slope of a line is the measure of the steepness and the direction of the line. Finding the slope of lines in a coordinate plane can help in predicting whether the lines are parallel, perpendicular, or none without actually using a compass.
The slope of any line can be calculated using any two distinct points lying on the line. The slope of a line formula calculates the ratio of the "vertical change" to the "horizontal change" between two distinct points on a line. In this article, we will understand the method to find the slope and its applications.
| 1. | What is Slope? |
| 2. | Slope of a Line |
| 3. | Slope of a Line Formula |
| 4. | How to Find Slope? |
| 5. | Types of Slope |
| 6. | Slope of Perpendicular Lines |
| 7. | Slope of Parallel Lines |
| 8. | FAQs on Slope |
What is Slope?
The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. The net change in y coordinate is Δy, while the net change in the x coordinate is Δx. So the change in y coordinate with respect to the change in x coordinate can be written as,
m = Δy/Δx
where,m is the slope
Note that tan θ = Δy/Δx
We also refer this tan θ to be the slope of the line.
Slope of a Line
The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane. Calculating the slope of a line is similar to finding the slope between two different points. In general, to find the slope of a line, we need to have the values of any two different coordinates on the line.
Slope Between Two Points
The slope of a line can be calculated using two points lying on the straight line. Given the coordinates of the two points, we can apply the slope of line formula. Let coordinates of those two points be,
P1 = (x1, y1)
P2 = (x2, y2)
As we discussed in the previous sections, the slope is the "change in y coordinate with respect to the change in x coordinate of that line". So, putting the values of Δy and Δx in the equation of slope, we know that:
Δy = y2 - y1
Δx = x2 - x1
Hence, using these values in a ratio, we get :
Slope = m = tan θ = (y2 - y1)/(x2 - x1)
where, m is the slope, and θ is the angle made by the line with the positive x-axis.
Slope of a Line Formula
The slope of a line can be calculated from the equation of the line. The general slope of a line formula is given as,
y = mx + b
where,
- m is the slope, such that m = tan θ = Δy/Δx
- θ is the angle made by the line with the positive x-axis
- Δy is the net change in y-axis
- Δx is the net change in x-axis
Slope of a Line Example
Let us recall the definition of slope of a line and try solving the example given below.
Example: What is the equation of a line whose slope is 1, and that passes through the point (-1, -5)?
Solution:
We know that if the slope is given as 1, then the value of m will be 1 in the general equation y = mx + b. So, we substitute the value of m as 1, and we get,
y = x + b
Now, we already have the value of one point on the line. So, we put the value of the point (-1, -5) in the equation y = x + b, and we get,
b = -4
Hence, substituting the values of m and b in the general equation, we get our final equation as y = x - 4.
Equation is: y = x - 4
How to Find Slope?
We can find the slope of the line using different methods. The first method to find the value of the slope is by using the equation is given as,
m = (y2 - y1)/(x2 - x1)
where, m is the slope of the line.
Also, the change in x is run and the change in y is rise or fall. Thus, we can also define a slope as, m = rise/run
Finding Slope from a Graph
While finding the slope of a line from the graph, one method is to directly apply the formula given the coordinates of two points lying on the line. Let's say the values of the coordinates of the two points are not given. So, we have another method as well to find the slope of the line. In this method, we try to find the tangent of the angle made by the line with the x-axis. Hence, we find the slope as given below.
The slope of a line has only one value. So, the slopes found by Methods 1 and 2 will be equal. In addition to that, let's say we are given the equation of a straight line. The general equation of a line can be given as,
y = mx + b
The value of the slope is given as m; hence the value of m gives the slope of any straight line.
The below-given steps can be followed to find the slope of a line such that the coordinates of two points lying on the line are: (2, 4), (1, 2)
- Step 1: Note the coordinates of the two points lying on the line, (x2, y2), (x1, y1). Here the coordinates are given as (2, 4), (1, 2).
- Step 2: Apply the slope of line formula, m = (y2 - y1)/(x2 - x1) = (4 - 2)/(2 - 1) = 2.
- Step 3: Therefore, the slope of the given line = 2.
Types of Slope
We can classify the slope into different types depending upon the relationship between the two variables x and y and thus the value of the gradient or slope of the line obtained. There are 4 different types of slopes, given as,
- Positive slope
- Negative slope
- Zero slope
- Undefined Slope
Positive Slope
Graphically, a positive slope indicates that while moving from left to right in the coordinate plane, the line rises, which also signifies that when x increases, so do y.
Negative Slope
Graphically, a negative slope indicates that while moving from left to right in the coordinate plane, the line falls, which also signifies that when x increases, y decreases.
Zero Slope
For a line with zero slope, the rise is zero, and thus applying the rise over run formula we get the slope of the line as zero.
Undefined Slope
For a line with an undefined slope, the value of the run is zero. The slope of a vertical line is undefined.
Slope of Horizontal Line
We know that, a horizontal line is a straight line that is parallel to the x-axis or is drawn from left to right or right to left in a coordinate plane. Therefore, the net change in the y-coordinates of the horizontal line is zero. The slope of a horizontal line can be given as,
Slope of a horizontal line, m = Δy/Δx = zero
Slope of Vertical Line
We know that, a vertical line is a straight line that is parallel to the y-axis or is drawn from top to bottom or bottom to top in a coordinate plane. Therefore, the net change in the x-coordinates of the vertical line is zero. The slope of a vertical line can be given as,
Slope of a vertical line, m = Δy/Δx = undefined
Slope of Perpendicular Lines
A set of perpendicular lines always has 90º angle between them. Let us suppose we have two perpendicular lines l1 and l2 in the coordinate plane, inclined at angle θ1 and θ2 respectively with the x-axis, such that the given angles follow the external angle theorem as, θ2 = θ1 + 90º.
Therefore, their slopes can be given as,
m1 = tan θ1
m2 = tan (θ1 + 90º) = - cot θ1
⇒ m1 × m2 = -1
Thus, the product of slopes of two perpendicular lines is equal to -1.
Slope of Parallel Lines
A set of parallel lines always have an equal angle of inclination. Let us suppose we have two parallel lines l1 and l2 in the coordinate plane, inclined at angle θ1 and θ2 respectively with the x-axis, such that the, θ2 = θ1.
Therefore, their slopes can be given as,
⇒ m1 = m2
Thus, the slopes of the two parallel lines are equal.
Important Notes on Slope:
- The slope of a line is the measure of the tangent of the angle made by the line with the x-axis.
- The slope is constant throughout a straight line.
- The slope-intercept form of a straight line can be given by y = mx + b
- The slope is represented by the letter m, and is given by, m = tan θ = (y2 - y1)/(x2 - x1)
Challenging Question:
A line has the equation y = 2x - 7. Find the equation of a line that is perpendicular to the given line, and is passing through the origin.
☛ Related Topics:
- Linear Equation
- Quadratic Equation
- Cubic Equation
FAQs on Slope
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. Slope can be calculated using different methods, given the equation of a line or the coordinates of points lying on the straight line.
What is the Formula to Find Slope of a Line?
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,
Slope = m = tan θ = (y2 - y1)/(x2 - x1)
How to Calculate the Slope?
The slope is found by measuring the tangent of the angle made by the line with the x-axis. There are different methods to find the slope of a line. The expression that can be used to find the slope is given as tan θ or (y2 - y1)/(x2 - x1), where θ is the angle which the line makes with the positive x-axis and (x1, y1) and (x2, y2) are the coordinates of the two points lying on the line.
What are the 4 Different Types of Slopes?
The 4 different types of slopes are positive slope, negative slope, zero slope, and undefined slope.
What is an Undefined Slope?
Any slope that has an angle of 90º with the x-axis, will have an undefined value of the tangent of 90º. Hence, such lines will have an undefined value of the slope.
What Does Slope Look Like?
The slope is nothing but the measure of the tangent of the angle made with the x-axis. Hence, it is just the measure of an angle.
What are 3 Ways to Find Slope?
The ways to find slope are point slope form, slope-intercept form, and the standard form. We can apply any of the forms of the equation of a straight line given the required information to find the slope.
How do you Show that Three Points are Collinear by Slope?
To prove the collinearity of three points, say A, B, and C, we can apply the slope formula. The slope of lines AB and BC should be equal for the three given points to be collinear points.
How to Find Slope With Two Points?
The slope can be calculated using the coordinates of two points using the formula, m = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points lying on the line.
FAQs
What is the slope formula example? ›
The slope intercept formula y = mx + b is used when you know the slope of the line to be examined and the point given is also the y intercept (0, b). In the formula, b represents the y value of the y intercept point. Example 2: Find the equation of the line that has a slope of 2/3 and a y intercept of (0, 4).
What are the 3 formulas for slope? ›...
| Fact | You can use this fact when you know: |
|---|---|
| Slope formula: | two points on a line |
| Slope-intercept formula: y = mx + b | the slope and y-intercept of a line |
| Point-slope formula: y - y1 = m(x - x1) | the slope of a line and a point on the line |
| Parallel lines have equal slopes | the slope of a line |
Slopes come in 4 different types: negative, positive, zero, and undefined. as x increases. The slope of a line can also be interpreted as the “average rate of change”.
What is the slope of a line example? ›Whenever the equation of a line is written in the form y = mx + b, it is called the slope-intercept form of the equation. The m is the slope of the line. And b is the b in the point that is the y-intercept (0, b). For example, for the equation y = 3x – 7, the slope is 3, and the y-intercept is (0, −7).
What is slope formula in 7th grade? ›SLOPE = Rise/Run= Change in Y over change in X. See the graphic below: The Rise is the difference (subtraction) between the y-coordinates of two points. The Run is the difference between the x-coordinates.
How do you find slope answer? ›- Pick two points on the line and determine their coordinates.
- Determine the difference in y-coordinates of these two points (rise).
- Determine the difference in x-coordinates for these two points (run).
- Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).
Some of the examples of linear equations are 2x – 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, 3x – y + z = 3.
What is the formula of a line? ›The equation of a straight line is y=mx+c y = m x + c m is the gradient and c is the height at which the line crosses the y -axis, also known as the y -intercept.
How do you define slope? ›What is slope? Slope is the steepness of a line as it moves from LEFT to RIGHT. Slope is the ratio of the rise, the vertical change, to the run, the horizontal change of a line.
What are the definitions of slope? ›1. : ground that forms a natural or artificial incline. : upward or downward slant or inclination or degree of slant.
What are the different types of lines? ›
How many types of lines are there? There are two basic lines in Geometry: straight and curved. Straight lines are further classifies into horizontal and vertical. Other types of lines are parallel lines, intersecting lines and perpendicular lines.
What is slope and y-intercept examples? ›In an equation in slope-intercept form (y=mx+b) the slope is m and the y-intercept is b. We can also rewrite certain equations to look more like slope-intercept form. For example, y=x can be rewritten as y=1x+0, so its slope is 1 and its y-intercept is 0.
What are the 4 types of equations? ›- Linear Equation.
- Radical Equation.
- Exponential Equation.
- Rational Equation.
A linear function is a function that represents a straight line on the coordinate plane. For example, y = 3x - 2 represents a straight line on a coordinate plane and hence it represents a linear function.
What are the 4 methods of solving linear equations? ›- The Graphing Method . ...
- The Substitution Method . ...
- The Linear Combination Method , aka The Addition Method , aka The Elimination Method. ...
- The Matrix Method .
- Find the slope using the slope formula. ...
- Use the slope and one of the points to solve for the y-intercept (b). ...
- Once you know the value for m and the value for b, you can plug these into the slope-intercept form of a line (y = mx + b) to get the equation for the line.
In geometry, a line is a straight one-dimensional figure that does not have a thickness, and it extends endlessly in both directions. Diagram A represents a line. It does not have any endpoint. The two arrows at each end signify that the line extends endlessly and is unending in both directions.
What is the equation formula? ›An equation is a mathematical statement that is made up of two expressions connected by an equal sign. For example, 3x – 5 = 16 is an equation. Solving this equation, we get the value of the variable x as x = 7.
Which is an example of a formula? ›Formula is an expression that calculates values in a cell or in a range of cells. For example, =A2+A2+A3+A4 is a formula that adds up the values in cells A2 through A4.
What is an example of a simple formula? ›For example, when you enter the formula =5+2*3, the last two numbers are multiplied and added to the first number to get the result. Following the standard order of mathematical operations, multiplication is performed before addition.
What is the example of basic formula? ›
1. Formulas. In Excel, a formula is an expression that operates on values in a range of cells or a cell. For example, =A1+A2+A3, which finds the sum of the range of values from cell A1 to cell A3.
What is slope and its types? ›Slope tells you how steep a line is, or how much y increases as x increases. Types of slopes. *postive slope (when lines go uphill from left to right) *negative slope (when lines go downhill from left to right) *zero slope (when lines are horizontal)
What are the 7 types of line? ›- Straight line.
- Curved line.
- Horizontal line.
- Vertical line.
- Parallel lines.
- Intersecting lines.
- Perpendicular lines.
- Transversal line.
There are many types of lines: thick, thin, horizontal, vertical, zigzag, diagonal, curly, curved, spiral, etc. and are often very expressive. Lines are basic tools for artists—though some artists show their lines more than others.
What are examples of y-intercept? ›When a linear equation is written in slope-intercept form (y=mx+b), the y-intercept is represented by the constant variable b. For example, in the equation y=6x+8, the variable b corresponds with 8. This is the y-intercept.
What is the slope given the points 3 5 and (- 2 4? ›Example: What is the slope of the line through the points (-2, 4) and (3, 5)? Therefore, the slope of the line is 1/5.
How do you find the slope of a line given a point? ›- Substitution method = plug in the slope and the (x, y) point values into y = mx + b, then solve for b. ...
- Point-slope form = y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the point given and m is the slope given.
Hence, we can find the slope with two variable by the formula m = ( y 2 – y 1 ) ( x 2 – x 1 ) .
How do you find a 2% slope? ›Slope can be calculated as a percentage which is calculated in much the same way as the gradient. Convert the rise and run to the same units and then divide the rise by the run. Multiply this number by 100 and you have the percentage slope.