chp 5 Gradient of a Straight Line

8/9/2019 chp 5 Gradient of a Straight Line

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Gradient of a Straight Line

Gradient of a Straight Line

The gradient of a straight line is the ratio of the verticaldistance to the horizontal distance between any two givenpoints on the straight line.

Gradient, m=Vertical distance Horizontal distance

Example:

Find the gradient of the straight line above.

Solution:


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Gradient of the Straight Line in Cartesian Coordinates

Finding the Gradient of a Straight Line

The gradient, m, of a straight line which passes through P ( x1, y1)

and Q ( x2, y2) is given by,

Example 1:


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Find the gradient of the straight line joining two points P and Q in the

above diagram.

Solution:

P ( x1, y1) (!, "), Q ( x2, y2) (1#, $)

%radient of the straight line PQ

Example 2:

&alculate the gradient of a straight line which passes through point A (', ")

and point B (", ).

Solution:


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A (*1, y1) (', "), B (*2, y2) (", )

%radient of the straight line AB

Intercepts

1. The x -intercept is the point of intersection of a straight line with the

xa*is.

2. The y-intercept is the point of intersection of a straight line with the

ya*is.

3. +n the above diagram, the xintercept of the straight line PQ is 6 and

the yintercept of PQ is .

!. +f the xintercept and yintercept of a straight line are given,


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Example 1

-hat is the xintercept of the line T/

Solution:

The xcoordinate for the point of intersection of the straight line with xa*is is #.!.

Therefore the xintercept of the line T is !.".

Example 2

Find the xintercept of the straight line 2 x 0 " y 0 #.

Solution:

2 x 0 " y 0 #

t xintercept, y #

2 x 0 "(#) 0 #

2 x

x "

Example 3

-hat is the yintercept of the straight line 12 x 1$ y #/

Solution:

12 x 1$ y #

t yintercept, x #

12(#) 1$ y #


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1$ y #

y !

Equation of a Straight Line

E"#ation of a Straight Line: y $ mx % c

1. %iven the value of the gradient, m, and the yintercept, c, an e3uation

of a straight line y mx 0 c can be formed.

2. +f the e3uation of a straight line is written in the form y mx 0 c, the

gradient, m, and the yintercept, c, can be determined directly from the

e3uation.

Example:%iven that the e3uation of a straight line is y " ! x. Find the gradient and y

intercept of the line/

Solution: y " ! x

y ! x 0 " 4 & y $ mx % c'

Therefore, gradient, m ! yintercept, c "

3. +f the e3uation of a straight line is written in the form ax 0 by 0 c #,

change it to the form y mx 0 c before finding the gradient and the

yintercept.

Example:

%iven that the e3uation of a straight line is ! x 0 y " #. -hat is the gradient

and yintercept of the line/

Solution:

! x 0 y " #

y ! x 0 "


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Equation of a Straight Line (Sample Questions)

Example 1

%iven that the e3uation of a straight line is ! x 0 y " #. -hat is the gradient of

the line/

Solution:

! x 0 y " #

y ! x 0 "y=#"$%&'%y=#('$&)(y=m$&cgradient, m=#('

Example 2

%iven that the e3uation of a straight line is y ' x 0 ". Find the yintercept of the

line/

Solution:

y m x 0 c, c is yintercept of the straight line.

Therefore for the straight line y ' x 0 ",

yintercept is "

Example 3


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Find the e3uation of the straight line 56 if its gradient is e3ual to ".

Solution:

%iven m "

ubstitute m " and (2, $) into y m x 0 c.

$ " (2) 0 c

$ 0 c

c 11

Therefore the e3uation of the straight line 56 is y $ 3 x % 11

(arallel Lines

&)' Gradient of parallel lines


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1. Two straight lines areparallel if they have

the same gradient.

+f PQ ** RS ,

then m PQ $ m RS

2. +f two straight lines have

the same gradient, then

they are parallel.

+f m AB $ mCD then AB ** CDExample 1:

7etermine whether the two straight lines are parallel.

&a' 2 y ! x =

y 2 x $

&+' 2 y " x !

" y 2 x 0 12


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&,' E"#ation of (arallel Lines

To find the e3uation of the straight line which passes through a given point

and parallel to another straight line, follow the steps below

Step 1 8et the e3uation of the straight line ta9e the form y mx 0 c.

Step 2 Find the gradient of the straight line from the e3uation of the

straight line parallel to it. Step 3 ubstitute the value of gradient, m, the xcoordinate and

ycoordinate of the given point into y = mx 0 c to find the value

of the yintercept, c. Step 4 -rite down the e3uation of the straight line in the form

y mx 0 c.

Example 2:Find the e3uation of the straight line that passes through the point (:, 2) and is

parallel to the straight line ! y 0 " x 12.


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Example 1

The straight lines 56 and ;< in the diagram above are parallel. Find the value

of q.


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chp 5 Gradient of a Straight Line

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