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Slope

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What is a Slope?

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The slope or gradient of a line specifies both the direction and steepness of a line. The slope is often represented by the letter ‘m’. The slope is determined by finding the ratio of “vertical change” to the “horizontal change” between any two different points on a line. The steepness or grade of a line is measured by the absolute value of the shape. A slope with a greater absolute value represents a steeper line. The direction of a line either rises, falls, and is horizontal or vertical.

  • A line that extends from left to right has a positive run and positive rise, and also yielding a positive slope i.e. m > 0

  • A line that declines from left to right has a negative run and negative fall, and also yielding a negative slope i.e. m < o

  • Horizontal lines have a zero positive slope, as they have zero rise and a positive run. 

  • The slope is undefined if the line is vertical as the vertical line has zero rise and any amount of run.

Slope Equation

As we know, Tan θ = \[\frac{Height}{Base}\]

And, we know that between any two given points 

\[\frac{Height}{Base}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

Finally, we get slope equations as: 

\[m=Tan\theta =\frac{\Delta y}{\Delta x}\]

Therefore, this becomes our final slope equation at any given point.


final slope equation at any given point


Equation of a Straight Line

The general equation of a straight line is represented in the form of y = mx + c, where m is the gradient and coordinates of the y-intercepts are (0,c). 

We can determine the equation of a straight line when the gradient and point on the line are given by using the formula: y - b = m ( x - a)

Here, m represents the gradients and (a, b) is on the line.


Equation of a Line Example

Q. Find the Equation of a Line With Gradient 5, Passing Through the Point ( 4,1).

Solution: 

Using the formula y - b = m ( x - a), and substituting the values: m = 5, a = 4, and b = 1

We get, y  - 1 = 5 ( x - 4)

y - 1 = 5x - 20

y =  5x - 20 + 1

y = 5x - 19

y =  5x - 19

Therefore, the equation of a line with gradient 5, passing through ( 4,1) is y =  5x - 19


How to Find Equation of Line Passing Through Two Given Points?

If a line passes  through two points M (x₁, y₁) and N (x₂, y₂) such that x₁ is not equal to x₂ and y₁ is not equal to y₂, the equation of a line can be found using the formula mentioned below:

\[\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\]

Here, the gradient (m) can be calculated as :

\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]


Line Perpendicular

Perpendicular lines are two lines that meet at right angles (90⁰). The slopes of the two lines are negative reciprocals of each other. This means when one line has a slope of m, a perpendicular line has a slope of -1/m.

When we get multiply slope m by perpendicular slope -1/m., we get the answer -1. 


What Does the Slope of a Velocity Time Graph Give?

Velocity is a term that measures both speed and direction of a moving body. A change in velocity is known as acceleration. When the velocity and time are graphed on the y - axis and the x-axis respectively, then the slope of the velocity-time graph gives the acceleration of the object.

The slope is the ratio of change in the y-axis and change in the x-axis.

Therefore, we can determine the slope by using the following formula:

\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

Here,

  • m represents the slope

  • y₂ - y₁ represents the difference in the unit on y-axis. 

  • x₂ - x₁ represents the difference in the unit on x-axis



Solved Example:

1. Find the Equation of a Line that Passes Through the Two Points (2,3) and ( 6,-5)

Solution:

The equation of a line through the point (2,3) and ( 6,-5) can be determined using the formula:


\[\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\]

As the gradient (m) is not given, we will find the gradient by using the formula:

\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

Substituting the values x₁ = 2, x₂ = 6, y₁ = 3, and y₂ = -5 in the above formula, we get,

\[m=\frac{-5-3}{6-2}\]

\[m=\frac{-8}{4}\]

m = -2

Using the formula y - y₁ = m (x - x₁), and substituting the values: m = -2 , x₁ = 2 and y₁  = 3

We get,  y - 3 = -2 ( x - 2)

y - 3 = -2x + 4

Therefore, the equation of a line passing through the point (-1,2) and ( 2,4) is 2x + y + 1 = 0.

2. Find the Equation of a Line That Passes Through the Point (2,0) and Has a Gradient -2.

Solution: 

Using the formula y - b = m ( x - a), and substituting the values: m = -2, a = 2 and b = 0

We get, y - 0 = - 2 (x - 2)

y - 0 = -2x + 4

y  =  -2x + 4.

FAQ (Frequently Asked Questions)

1. What is a Zero Slope Line Equation?

Ans. A zero slope line is a perfectly flat, and straight-line running along the horizontal axis of a cartesian plane. The equation of a zero line is one where the x values changes but the y values remain constant. Therefore, the equation for a zero line is given as y = b, where the slope of the line is m (m = 0).

2. What Does the Negative Slope Mean?

Ans. The negative slope means that the relation of two variables is negatively related i.e. when the value of the variable x increases, the value of the variable y decreases. Graphically, the negative slope of a line means as the line moves from left position to right position, the line falls.

3. What Does the Positive Slope Mean?

Ans. The positive slope means that the relation of two variables is positively related i.e. when the value of the variable x increases, the value of the variable y also increases, and vice versa. Graphically, the negative slope of a line means as the line moves from the right position to left position, the line rises.