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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