Introduction
Welcoming students to this insightful guide, we will explore how to find the slope of straight lines. Understanding the concept of slope is fundamental in geometry and algebra, serving as a vital tool in various real-life applications, including architecture and engineering. This guide will cover key concepts, including formulas for finding slopes, properties of parallel and perpendicular lines, and step-by-step examples.
Understanding Slope
What is Slope?
The slope of a line represents its steepness and direction. Mathematically, it is defined as the change in the vertical distance (y) divided by the change in the horizontal distance (x). The formula can be summarized as:
[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} ]
Where:
- ( m ) is the slope,
- ( (x_1, y_1) ) and ( (x_2, y_2) ) are two points on the line.
Coefficients in the Line Equation
The general equation of a straight line can be expressed in the slope-intercept form as:
[ y = mx + c ]
Where:
- ( m ) is the slope,
- ( c ) is the y-intercept (the value of y when x = 0).
In our examples, we'll frequently reference coefficients, with slopes often derived from equations written in forms set to zero.
Finding the Slope of a Line
Using Coefficients
To find the slope using the coefficients in an equation, let's consider an example:
- Given Equation: ( 7x - 3y + 17 = 0 )
- Rearranging gives us ( -3y = -7x - 17 ) or ( y = \frac{7}{3}x + \frac{17}{3} )
- From this, we identify the slope ( m = \frac{7}{3} ).
- Parallel Lines: Refers to lines having the same slope. For instance, two lines with slopes ( \\ m_1 = m_2 = \frac{7}{3} ) are parallel.
- Perpendicular Lines: The slopes of perpendicular lines are negative reciprocals. For a slope ( m_1 ), the perpendicular slope ( m_2 ) will be:
[ m_1 imes m_2 = -1 ]
If ( m_1 = \frac{7}{3} ), then ( m_2 = -\frac{3}{7} ). - Undefined Slopes: When the slope calculation results in division by zero, the slope is undefined, representing a vertical line.
Application Example
Let’s take a more complex example of finding slopes using point coordinates:
- Find the slope of the line through points (2, 3) and (5, 11):
- Slope ( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 3}{5 - 2} = \frac{8}{3} )
- Thus, the slope is ( \frac{8}{3} ).
Common Slope Problems
Collating Knowledge
- Parallel Lines: ( y = 4x + 2 ) and ( 2y = 8x + 4 ) have slopes of 4, confirming they are parallel.
- Perpendicular Lines: For the line ( x + 4y = 16 ), transforming it provides: ( y = -\frac{1}{4}x + 4 ), indicating a slope of -1/4, showing if it is perpendicular to another with slope 4.
Concluding Thoughts
Finding the slope of straight lines is essential in understanding linear relationships in mathematics. We’ve covered basic definitions, derived slopes from equations, and assessed parallelism and perpendicularity between lines through real-life examples.
Understanding these concepts builds a solid foundation for further exploration into calculus and advanced mathematical topics. With practice, calculating slopes will become a straightforward task for any student.
Thank you for engaging with this guide! Feel free to reach out with questions or for clarification on topics covered.
hello students welcome to OnePlus One Channel 10 standard chapter 5 exercise
5.4o in the probably patina find the slope of the following straight lines equal to MX plus C and the formula is
[Music] which is equal to m is equal to minus
coefficient of x by coefficient of
y so in the formula you're spinning number director alterations next to some paranga
7x minus 3 by 17 is equal to 0 or the sum m is equal to
minus coefficient of x by coefficient of
y okay if a number value substitute minus coefficient of x x 7 by coefficient of y y it is understood
any number by zero is nothing but not defined that means Infinity not defined not Define
undefined okay so the summer slope when they're not defined very simple concept the formulas
find the slope of the line which is parallel to Y is equal to 0.7 x minus 11 and perpendicular to the line X is equal
to minus 11. [Music] find the slope of the line which is
foreign coefficient of x by
coefficient of white foreign another
parallel slope is nothing but same same either no change perpendicular slope changes
perpendicular slope perpendicular slope is
nothing but it is infinity Zero by minus one zero by minus one another zero okay Zero by minus one zero
by minus one another zero either down and this slope of this sum foreign foreign
foreign slope of straight line
slope of straight lines X by coefficient of
y term s one by four okay
this implies straight line to other M2 minus coefficient of x 1 the 2 by 3 2 by
3 by coefficient of y1 by 2 so which is equal to minus 2 by 3 by 1 row this is nothing but minus 2
into 2 will the minus 4 by 3 into 1 3 so in the samla patina M1 is equal to M two other slope one is
equal to slope two check whether the given lines are parallel or perpendicular therefore the given lines
are given lines are another parallel random equals
perpendicular okay surrender slope May equal updated the given lines are parallel okay
indications coefficient of x by coefficient of Y is x minus coefficient of x by coefficient
of Y minus 5 minus minus cancel either for 23 by 5 is therefore the given lines are
the given lines are perpendicular simple Concepts equal to minus 1 and 1 over
foreign [Music] foreign
slope of straight line so equations
coefficient of X by coefficient of Y minus coefficient of x
patina P plus 3. by coefficient of Y patina 12 . number brackets
12x minus seven Y is equal to 16. minus 16 which is equal to zero upon a slope of
straight line minus coefficient of x 12 by coefficient of Y minus 7 minus minus cancel out of
12 by 7 when the M2 E4 in the straight line of perpendicular solution
since the given straight lines are perpendicular
since the given straight lines are perpendicular it is understood M1 is equal to m 2 is equal M1 into M2 is
equal to minus 1. even a value substitute minus of p plus 3 by 12 into m 2 1 the 12 by 7 is equal to minus 1
and 4 the question is [Music] minus 3 which is nothing but p is equal
to 4 p over the value is foreign foreign
passing through the point minus 5 comma 2 and parallel to the line joining the points C comma minus 2 and minus 5 comma
4. otherwise foreign and parallel to the line joining the
points in the equation on the straight line when the in the render pointer joints
is slope point formula for state of decline other slope formula Y2 minus y1 by x 2
minus x 1 equal number this slope of QR slope of q r is equal to the X1 y1 X2 Y2 director
by 2 minus 4 formula minus y 1 core minus minus 2 when the minus the minus
x 2 minus X1 upper minus 5 minus three four plus two one this six by minus 5 minus 3 with the minus 8 okay if I
cancel particular number two tabular two threes are two fours minus three by four and the minus
foreign minus eight is equal to minus three x minus 50.
foreign [Music] so equation of the straight line number
control 3x plus 4y plus 7 is equal to 0. [Music] find the equation of a line passing
through six comma minus 2 and perpendicular to the line joining the point 6 comma 7 and 2 comma minus
find the equation of a line passing through six comma minus 2 and perpendicular to the line joining
the points line in the render Point here join fundraise
6 comma minus two in the point where yeah it is understood
of A and B can duplicate allow the six comma 7 and 2 comma minus three
questions minus 3 minus 7 minus 3 minus 7 by 2 minus 6 2 minus 6. minus 3 minus 7 when
the minus 10 by 2 minus 6 on the minus 4 minus minus cancel Panic for slope
five by two EPO in the point and the point and the C six comma minus 2 by U and the slope
which is perpendicular slope foreign
foreign equation of a straight line Y minus y 1 is equal to M into x minus x 1 so is the
X1 y1 so y minus y 1 over minus 2 and understand the plus side of Y plus 2 m 1 the minus 2 by 5 into x minus x 1 1 is
6. foreign plus 2X plus 10
minus 12 is equal to 0.0 degrees minus 2 is equal to
0.1 the equation of a straight line perpendicularly is okay
foreign hmm 7th is
C are the vertices of triangle ABC find the equation of the altitude through a and b b
triangle dropper easy understand find the equation of the altitude through a and
BT altitude is nothing but perpendicularly altitude
and equation of b e equation foreign
x 2 minus x 1 the slope for the formula substitute y two three formula minus y1 minus 2 so
minus already formula or minus 2 by x 2 12 minus x 1 1 the 10. 5 to 3 plus 2 on the 5 by 12 minus ten one the
two so if I'm on the slope of BC when the 5 by 2 will continue on the slope of altitude BC altitude
slope of altitude um slope altitude altitude
foreign x minus X1 is equal to sorry y minus y1 for a point or stop
form y minus y 1 is equal to M into x minus X1 in the formula version among the equation
X1 y1 would I substitute y minus y one one is zero is equal to M1 minus 2 by 5 into x minus x 1 with the
minus three so I'll do your minus 0 plus side so Plus 3. foreign
equation of altitude number equation of altitude in a
question is slope of AC Y2 minus y1 by X2 minus X1 so an update
directory so Y2 into 3 minus y1 0 by X2 Under 12 minus X1 with the minus 3 upper random
minus in the plus side three minus 0 3 by 12 plus 3 1 to 15. another slope of the altitude
slope of the altitude is B even there minus 5.
equation of altitude equation of altitude B
point B point when the 10 comma minus two sorry ten comma minus 2 slope under
foreign y1 y minus y1 with the minus 2 minus so
minus in the plus side of plus 2 is equal to M when the minus 5 into x minus x 1 10.
y plus 2 is equal to is minus 0 minus plus 5 tens are 50. foreign
equation of altitude a d is equation of altitude
b e is 5X plus y minus 48 is equal to 0.80 2x plus 5y
2x plus 5y plus 6 is equal to 0. okay foreign joining the points
foreign foreign foreign
slope of a b formula Y2 minus y1 by x 2 minus X1 so the X1 y1 X2 Y2 number substitute Y2 and
the minus 4 formula minus y 1 1 to 2 by x 2 1 the 6 formula minus x 1 1 minus 4 already or minus in the plus side of
Plus 4. minus 4 minus 2 and the minus 6 by 6 plus 4 1 the ten if I cancel three two uh three twos are six two five are
ten upper minus three by five foreign CD
allow the perpendicular pan is foreign y1 plus y two by two so the X1 y1 either
x two either by 2 so substitute x 1 1 there minus 4 plus x 2 1 to 6 by 2 comma y one one the two plus y two and the
minus 4 something or minus four one by two foreign
so equation of the perpendicular bisector CD about this seeing at a point when the
one comma minus one seeing the point Sorry dingra point one
there D 0.1 comma minus 1. y minus
y1 is equal to M into x minus X1 so related to the X1 y1 so y minus y1 with the minus 1 soliding or minus Circle and
minus in the plus side of Y minus 1 equal to m 1 to 5 by 3 into x minus x 1 1 to 1.
minus 1. minus five foreign
0 answer 5x minus 3 by minus 8 is equal to 0 the answer is
equation of the uh straight line numbers foreign intersection of this lines
intersection intersection on our Point foreign Apple
solving equations elimination methods solving equation
7x plus 3y equal to 10 and 5x minus 4y is equal to 1 either foreign
foreign foreign [Music]
is very good either seven on the seven fives are thirty five five sevens are 35 people is X7 the number of same markets
eliminate the foreign 28 plus 15.
free balance 1 3 43 x is equal to India 43 from Mexico so the 43 in the column of the Divide level so 43 by 43
cancellator x is equal to 1. even number x value conductor is Y is equal to one
one subscribe one comma one
line thirteen X plus five y plus 12 is equal to 0.
so line parallel to 13x plus 5y plus 12 yes
foreign equation of the straight line is 13x plus 5 by okay
minus 18 is equal to 0. formula y minus y 1 is equal to M into x minus X1
making a corner slope of equation slope of the equation minus
coefficient of x by coefficient of like formula minus coefficient of x 1 to 13 by
coefficient of y u and the Phi if a point slope form for the equation going to be no y minus y1 is equal to M into x
minus x 1.1 comma 1 so y1 y minus 1 is equal to m 1 the minus 3 13
by 5 x minus x 1 1 5 into y minus 1 is equal to minus 13 into x minus 1 multiplier five y minus 5 is
equal to minus 13x minus minus plus 30. Yeah couple five eyes minus 5 minus 13 plus 13x minus plus 13 minus 13 is equal
to 0.13x Minus 5 minus 13 when the minus 18 is
equal to 0. is foreign
so find the equation of a straight line through the intersection of this line my intersecting points
five x minus 6y is equal to 2 now equation 1 and what's current 3x plus 2y is equal to 10 equation two in which
the equation is so solving equation 1 and 2. foreign um
9x three twos are six Y is equal to 30. you pay the cancel add-on 5x plus 9 x 1
through 14x is equal to 32 number X is equal to 32 by 14 so the cancel panicama two tables
X is equal to 16 by 7 so substitute 16 by 7 in equation is
equation two one two three X plus two Y is equal to 10 in the equation 16 by 7 plus 2y is equal to ten so the
multiply Panna 3 into 16 48 row 48 by 7 plus 2y is equal to 10 at the numbers
for 48 plus 14 by by 7 0 is equal to 10 so 48 plus 14 Y is equal to 17 um
10 minus 8 1 the 2 6 minus 4 on the 22 Verma yes
if a y is equal to 22 by 14 2 table a cancel Panic Eleven by seven value
the intersection Point number control [Music] intersecting point one there
sixteen comma 7 16 by 7 comma Eleven by seven if a perpendicular equation perpendicularly
perpendicular to the line
another 4X minus 7y plus 13 is equal to zero in the perpendicular equation in the coefficient of the exchange in
the seven coefficients so passes through passes through 16 comma 7 11 sorry 16 by
7 11 by 7 in x equals 16 by seven eleven by seven points for seven into sixteen by seven plus four into Eleven by seven
plus K is equal to 0 another value control the equation of the straight line
sleeping is seven seven cancels sixteen Plus 44 by 7 plus K is equal to zero
language cancel foreign Verma
[Music] 156 okay 156 plus 7K equal to 0 okay
7K is equal to minus 156 equal to minus 156 by 7. 7 cancels substitute equation of
a straight line straight line one then seven X plus 4y plus k equal to 0 the straight line
equation of a 7x plus 4 y k over the value on the minus 156 by 7 is equal to 0. when the 7 is
156 is equal to zero so it is foreign foreign
joining the intersection of these two two straight lines equation to the point of intersection of these two straight
lines to the point of intersection of this two straight lines
foreign simple concept so if an eye equation is
first three X Plus y plus 2 is equal to zero okay the equation one net x minus 2y minus 4 is equal to 0 is the
equation foreign foreign
foreign minus one is foreign
solving equation 1 and 2 eliminate eliminate
upon first equation well I'm sorry six six three twos are six
two y two twos are four is equal to zero full equation second equation
x minus 2y minus 4 is equal to zero so the cancel it cancel out of four minus four zero six X Plus One X seven X seven
X is equal to zero up x what a value one then Zero by seven no row of X order value one is zero
foreign point of intersection
or a point of intersection 0 comma minus two so yeah seven x minus 3 Y is equal to minus 12
2y equal to X Plus 3. third equation fourth equation is equal to zero foreign
foreign foreign equation
3 and 4 eliminate X is equal third equation
third equation 7X minus 3y plus 12 is equal to zero fourth equation of the number seven I'll
multiply 7X 14 y 21 7x minus 14 y 7 3 is our 21 is equal to
0. 7 x 7 x cancels minus minus i j minus 10 plus size minus signs
you know cancel minus three plus fourteen eleven 12 minus 21
minus plus sorry minus nine is
equal to 9 by 11. so at the point of intersections substitute Y is equal to 9 by 11
in equation third equation 7x minus 3y plus 12 is equal to 0. 7x minus 3 into Yoda value
in the 9 by 11 plus 12 is equal to 0. super 7x 9 3 is at 27 27 by 11 plus 12 is equal to 0.
foreign foreign foreign
so the equation of a straight line conduct two point form two point in a bit of the class first two point one
this zero comma minus two zero comma minus two other minus 15 comma eleven nine comma eleven two point
four so even the X1 y1 you know the X2 Y2 upper two point form equation of a straight line
X2 minus x 1 is equal to Y minus y 1 by Y2 minus y 1 substitute m x minus x 1 1 is 0 by x 2 minus 15 by 11 minus x 1 1
the 0 is equal to Y minus y1 when the minus 2 already over minus plus side of plus 2 by Y2 on the 9 by 11
minus y 1 will 2 plus 2 again in the summer X by
minus 15 by 11 is equal to Y plus sorry y plus 2 by 9 by 11 plus 2 up in the 11th
9 plus 22 by 11.9 plus 22 by 11 first denominator simply
minus 15 by 11 is equal to y plus 2 Z by at the 22 plus 9 1 the 31 by 11
keep a cross multiply it is X into 31 by 11 is equal to Y plus 2 into minus 15 by 11. Illinois
foreign foreign foreign
find the equation of a straight line through the point of intersection of the lines and bisecting the line segment
joining the points of intersection and bisecting the line segment joining
the points in the render line in the render pointer lineup
it is understood in the render equation you first is foreign
equation 1 and 2 eliminate X first equation
eight X plus 3y is equal to 18 second equation
photos are eight x five twos are ten by nine twos are eighteen Zero by minus seven zero
you know so substitute Y is equal to 0
in equation is eight X Plus 3y is equal to 18 by 8 X plus 3 minus 3 into 0 is equal to 18 in
the term full of zero it is 8x is equal to 18 up X is equal to 18 by 18 x is equal to two times
so point of intersection point of intersection when the first x value nine by four add two by your value
0. minus seven comma six one y One X two y two
midpoint formulas X1 plus x 2 by 2 comma y1 plus Y 2 by 2.
x 1 1 5 plus x 2 and the minus 7 upper minus seven okay by 2 y 1 1 minus 4 plus Y 2 1 the 6 by
2. minus seven when the minus 2 by 2 comma minus 4 plus 6 1 to 2 by 2 so the cancel
Adder one dollar of a minus one comma one foreign equation of straight line
first point nine by four comma zero minus one comma 1. x 1 y 1 x 2 Y2 so
formula patina two point formula by x minus x 1 by x two minus x 1 is equal to Y minus y1 by Y2 minus y one is
so x minus x 1 1 there 9 by 4 fraction number x two on the minus 1 minus x 1 over 9 by
4 is equal to one so upon the Y minus y1 0 by Y2 and the 1 minus y 1 1 is
0.4 x minus nine by four and one bye again minus four minus 9 by 4. y minus zero Y
is 4X minus nine by four over by minus four minus nine patina minus thirteen by four
is equal to Y minus one in every cross multiply Panic line is foreign
minus nine is equal to 0. foreign foreign so thank you so much for watching
students bye bye
Heads up!
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