How To Solve: Slope of a Line

Attached pdf of this Article as SPOILER at the top! Happy learning!

Hi All,

I have recently uploaded a video on YouTube to discuss Slope of a Line in Detail:




Following is covered in the video

¤ Proof of Slope of a Line formula
¤ How to find slope of a Generic Line
¤ Sign of Slope of a Line
¤ Slope of Parallel and Perpendicular Lines
¤ Solved Problem : Find slope of a line given two points
¤ Solved Problem : Find slope of a line given equation of a line
¤ Solved Problem : Find value of a variable given the slope of a line
¤ Solved Problem : Find equation of a line parallel to a line and passing through a point
¤ Solved Problem : Find equation of a line perpendicular to a line and passing through a point

Proof of Slope of a Line formula

Let's say we have a line(l) passing through two point A(\(x_1\),\(y_1\)) and B(\(x_2\),\(y_2\)) and having a slope of m.
Lets draw perpendicular lines as shown in the below figure to complete △ABC
Let ∠BAC = θ



Slope of a line, m = tan(Angle made by the line with positive X-Axis) = tan(θ) = Opposite Side / Adjacent Side = \(\frac{BC}{AC}\) = \(\frac{y_2 - y_1}{x_2 - x_1}\)


How to find slope of a Generic Line

Let's say we have a generic equation of the line given by ax + by = c
=> by = -ax + c
=> y = \(\frac{-a}{b}\)x + \(\frac{c}{b}\)

Comparing above equation with generic equation of a line y = mx + b we get
m = \(\frac{-a}{b}\)

=> Slope of the line ax + by = c, is given by m = \(\frac{-a}{b}\) = -Coefficient of x / Coefficient of y


Sign of Slope of a Line

¤ Positive Slope: Line tilted towards right
¤ Negative Slope: Line tilted towards left
¤ Zero Slope: Line parallel to x-axis
¤ Infinite Slope: Line parallel to y-axis




Slope of Parallel and Perpendicular Lines

If two lines are Parallel, then their slopes will be equal.

¤ If we have two parallel lines \(L_1\) and \(L_2\), with below equations
¤ Line L1 : y = \(m_1\)x + \(c_1\)
¤ Line L2 : y = \(m_2\)x + \(c_2\)
¤ then \(m_1\) = \(m_2\)



If two lines are Perpendicular, then product of their slopes will be equal to -1

¤ If we have two perpendicular lines \(L_1\) and \(L_2\), with below equations
¤ Line L1 : y = \(m_1\)x + \(c_1\)
¤ Line L2 : y = \(m_2\)x + \(c_2\)
¤ then \(m_1\) * \(m_2\) = -1




Solved Problem : Find slope of a line given two points

Q1. Find the slope of the line passing through two points (1,3) and (3,5).




Solved Problem : Find slope of a line given equation of a line

Q2. Find the slope of the line 2x + 3y = 5.




Solved Problem : Find value of a variable given the slope of a line

Q3. A line with slope 2 passes through the points (x,5) and (2,x). Find the value of x.




Solved Problem : Find equation of a line parallel to a line and passing through a point

Q4. Find the equation of the line which is parallel to the line 2x + 4y = 5 and passes through the point (2,3).




Solved Problem : Find equation of a line perpendicular to a line and passing through a point

Q5. Find the equation of the line which is perpendicular to the line 2x + 4y = 5 and passes through the point (2,3).



Watch the following video to learn the Basics of Co-ordinate Geometry



Watch the following video to learn about Equation of a Line



Hope it helps!
Good Luck!