GRE Prep Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Moderator
Joined: 02 Jan 2020
Status:GRE Quant Tutor
Posts: 1111
Given Kudos: 9
Location: India
Concentration: General Management
Schools: XLRI Jamshedpur, India - Class of 2014
GMAT 1: 700 Q51 V31
GPA: 2.8
WE:Engineering (Computer Software)
How To Solve: Slope of a Line
[#permalink]
15 May 2023, 09:24
15 May 2023, 09:24
1
Show: :: DOWNLOAD PDF: How To Solve: Slope of a Line
Attachment:
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
Attachment:
Slope of a Line - 1.jpg [ 16.96 KiB | Viewed 1568 times ]
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
Attachment:
Sign of slope.jpg [ 22.08 KiB | Viewed 1589 times ]
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 \(m_1\) = \(m_2\)
Attachment:
Parallel Lines.jpg [ 4.93 KiB | Viewed 1560 times ]
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
Attachment:
Perpendicular Lines.jpg [ 3.41 KiB | Viewed 1542 times ]
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).
Show: ::
Sol 1: Slope of a line is given by m = \(\frac{y_2 - y_1}{x_2 - x_1}\) = \(\frac{5 - 3}{3 - 1}\)
= \(\frac{2}{2}\) = 1
= \(\frac{2}{2}\) = 1
Solved Problem : Find slope of a line given equation of a line
Q2. Find the slope of the line 2x + 3y = 5.
Show: ::
Sol 2: Slope of the line ax + by = c, is given by m = -Coefficient of x / Coefficient of y
= \(\frac{-2}{3}\)
= \(\frac{-2}{3}\)
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.
Show: ::
Sol 3: Slope of a line is given by m = \(\frac{y_2 - y_1}{x_2 - x_1}\)
= \(\frac{x - 5}{2 - x}\) = 2 (given)
=> x - 5 = 4 - 2x
=> 3x = 9
=> x = \(\frac{9}{3}\) = 3
= \(\frac{x - 5}{2 - x}\) = 2 (given)
=> x - 5 = 4 - 2x
=> 3x = 9
=> x = \(\frac{9}{3}\) = 3
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).
Show: ::
Sol 4: Equation of line 2x + 4y = 5 = \(\frac{-2}{4}\) = \(\frac{-1}{2}\)
=> Parallel line will have the same slope = \(\frac{-1}{2}\)
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \(\frac{-1}{2}\) * (x - 2)
=> 2y - 6 = -x + 2
=> x + 2y = 8 is the equation of the line parallel to 2x + 4y = 5 and passes through the point (2,3)
=> Parallel line will have the same slope = \(\frac{-1}{2}\)
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \(\frac{-1}{2}\) * (x - 2)
=> 2y - 6 = -x + 2
=> x + 2y = 8 is the equation of the line parallel to 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).
Show: ::
Sol 5: Equation of line 2x + 4y = 5 = \(\frac{-2}{4}\) = \(\frac{-1}{2}\)
=> Slope of perpendicular line, m will be given by m * \(\frac{-1}{2}\) = -1
=> m = 2
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \([fraction]2[/fraction]\) * (x - 2)
=> y - 3 = 2x - 4
=> y = 2x - 1 is the equation of the line perpendicular to 2x + 4y = 5 and passes through the point (2,3)
=> Slope of perpendicular line, m will be given by m * \(\frac{-1}{2}\) = -1
=> m = 2
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \([fraction]2[/fraction]\) * (x - 2)
=> y - 3 = 2x - 4
=> y = 2x - 1 is the equation of the line perpendicular to 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!
gmatclubot
How To Solve: Slope of a Line [#permalink]
15 May 2023, 09:24
Moderators:
|
|
||
