Slope of a line: Definition, Formulas and Examples

A Slope of a Line is defined as the ratio of rise over run that is change in “y” divided by change in “x”. In simple terms, slope of a line is a measure of its steepness.

\(Slope=\frac{rise}{run}=\frac{\Delta y}{\Delta x}\)

Slope of a line is generally denoted by “m”. In geometry, we have seen lines in a coordinate plane. To understand whether these lines are parallel, perpendicular or at a certain angle without using geometrical tools, the best way is by measuring its slope.

Read more: Polygon curve angle

Key Terms: Slope, Parallel Line, Perpendicular Line, Collinear Points, y-intercept.

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What is a Slope?

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Slope is basically the measure of how steep a straight line is. It is denoted by “m” and measured by the change in value of y (Δy) divided by the change in value of x (Δx).

Hence, the equation of slope of a line can be written as,

m = \(\frac{change in y}{change in x}\)

Or, m =\(\frac{\Delta y}{\Delta x}\)

The slope of a line can also be represented by “tan θ”.

Therefore, the equation of slope can also be written as,

tan θ = \(\frac{\Delta y}{\Delta x}\)

Hence, the slope of a straight line formed by two points P and Q with coordinates (x₁ , y₁) and (x₂ , y₂) respectively can be determined by the difference of the coordinates of the points P and Q.

The formula of slope thus formed will be,

m = \(\frac{y_2- y_1}{x_2- x_1}\)

Based on the above formula, we can easily determine the slope of a line between two points.

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Equation of a Line

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Equation of a Line is a mathematical expression that shows the relation between the coordinate points lying on a straight line. The equation of a straight line are of 3 types namely:

• Standard Form: ax+by=c where a, b, c are constants and x and y are variables.
• Slope-Intercept Form: y = mx + c where m is the slope and c is the y-intercept.
• Point-Slope Form: y-y1=m(x-x₁) where (x, y) is an arbitrary point on the line and (x₁, y₁) is a known point on the line. 

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Slope of Parallel Lines

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Slope of two lines which are parallel to each other is always equal since parallel lines are equally inclined or steep. If two lines l₁ and l₂ who are parallel to each other have slopes m₁ and m₂ respectively, then we have,

m₁ = m₂

Derivation:

Let, two lines l₁ and l₂ have slopes m₁and m₂ respectively forms an angle “θ” between them.

Therefore,

tan = \(\frac{m_2- m_1}{1+ m_1m_2}\)

Since the lines are parallel, the angle between them is 0˚ or 180˚.

Thus, tan 0˚ = tan 180˚ = 0,

Therefore we get,

0= \(\frac{m_2- m_1}{1+ m_1m_2}\)

Or, 0 = m₂ – m₁

Or, m₁ = m₂ [Derived]

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Slope of Perpendicular Lines

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Slope of two lines that are perpendicular to each other is such that the slope of one line is the negative reciprocal of the slope of another line.

In simple terms, the product of the slope of two perpendicular angles is equal to -1.

Therefore, two lines l₁ and l₂ having slopes m₁ and m₂ respectively and are perpendicular to each other can be represented as,

m₁ . m₂= – 1

Derivation:

The above figure shows two perpendicular lines l₁ and l₂ with inclinations α and β.

Since they are perpendicular, we can say that β = α + 90˚ [using properties of angles].

Therefore their slopes can be represented as,

m₁ = tan (α + 90°) and m₂= tan .

Or, m₁ = – cot cot α

Or, m₁= – \(\frac{1}{tantan}\) = – \(\frac{1}{m_2}\)

Or, m₁× m₂ = – 1 [Derived]

Thus, two lines are perpendicular to each other only if their slopes are negative reciprocals of each other.

Read more: 

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Slope for Collinearity

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For three points A, B and C to be collinear, the slope of lines formed by any two points (say AB and BC) should be equal and there should be at least one common point through which they should pass. In other words, three points A, B and C will be collinear only if,

Slope of line AB = Slope of line BC = Slope of line AC

The slope of each line can be determined by the equation,

m = \(\frac{y_2- y_1}{x_2- x_1}\)

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Angle between Two Lines

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Angle between Two Lines

Let L₁ and L₂ be two lines with slopes m₁ and m₂ respectively. The point of intersection created a pair of vertically opposite angles and a pair of adjacent angles. If α₁ and α₂ are the inclinations of the two lines L₁ and L₂ respectively, then,

m₁ = tan ₁ and m₂ = tan₂

Let, θ and ɸ be the adjacent angles of L₁ and L₂.

Θ = α₂ – α₁ and α₁, α₂ ≠ 90˚.

Therefore,

tanθ = tan tan (α₂ – α₁) = \(\frac{tan \alpha_2-tan\alpha_1}{1+tan\alpha_1\alpha_2}\) = \(\frac{m_2- m_1}{1+ m_1m_2}\)

As, 1 + m₁m₂ ≠ 0 and ɸ = 180˚ – θ,

tanɸ = tan (180˚ – θ) = – tanθ = – \(\frac{m_2- m_1}{1+ m_1m_2}\)

Here two cases arise.

1. If \(\frac{m_2- m_1}{1+ m_1m_2}\) is positive, the tanθ will be positive and tanɸ will be negative which determines that θ is acute and ɸ is obtuse.
2. If \(\frac{m_2- m_1}{1+ m_1m_2}\) is negative, then tanθ will be negative and tanɸ will be positive which determines that θ is obtuse and ɸ is acute.

Thus the acute angle θ can be found by,

tan tan θ = \(\frac{m_2- m_1}{1+ m_1m_2}\)

And the obtuse angle ɸ can be found by,

ɸ = 180˚ – θ.

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Things to Remember

1. Using two points of a line to calculate its slope.
2. By picking different points on a line you can double check your answer.
3. If the line is going up from left to right, the slope of the line is positive and if it is going down from left to right, the slope of the line is negative.
4. There are 4 types of slope: Positive, Negative, Zero and Undefined.
5. A horizontal line has a slope 0 whereas a vertical line has an undefined slope.
6. Equation of a line passing through the origin is y=mx.