The Straight Line - Slide

7/30/2019 The Straight Line - Slide

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MATHEMATICS FORM 4

CHAPTER 5 : THE STRAIGHT LINE

SUBTOPIC 5.4 : EQUATION OF A

STRAIGHT LINE


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GRADIENT

vertical distancemhorizontal distance

2 1

2 1

m

y y

x x

y interceptm x intercept


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vertical distancem

horizontal distance

Vertical distance

horizontal distance

y

x


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2 1 1 2

2 1 1 2

y y y ymx x x x

y

x

( x2,y2 )

( x1,y1 )


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y interceptmx intercept

y

x

x-intercept

y-intercept


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WRITING THE EQUATION OF STRAIGHT LINE

In the equation of straight liney=mx + c, m is the gradient and c isthey-intercept.

Ex:Write equation of straight line, giventhe gradient, m and they-intercept, c.

a) Gradient = 4,y-intercept = 6

b) Gradient = -2,y-intercept = -7


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Solution

a) By substitute m= 4 and c = 6 into the

equationy=mx + c

Hence,

The equation of the straight line

y = 4x + 6


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Solution..

b) Substitute m= -2, c = -7 into the

equationy=mx + c.

Hence,

The equation of straight line is

y=-2x -7


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GRADIENT AND y-INTERCEPT

If the equation of the straight line is given in

the formy=mx + c, then the coefficient ofx,m, is the gradient whereas the constant, c, isthey-intercept.

Ex:

State the gradient andy-intercept of thestraight lines represented by the followingequations.

a) y = 9x -1

b) y = -4x +10


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Solution

a) y = 9x

1Compare to they=mx + c where m is gradient

and c isy-intercept.

Hence,Gradient = 9

y-intercept = -1


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Solution..

b) y = -4x + 10

Compare to they=mx + c where m is gradient

and c isy-intercept.Hence,

Gradient = -4

y-intercept = 10


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If the equation of a straight line in the

form ofpx + qy = r, then we need tochange the subject of the formula toy

so that it is in the form ofy = mx + c.

The gradient andy-intercept can beeasily determined.


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Ex:

Determine the gradient and y-intercept

of each of the following straight lines.

a) 6x + 3y = 8

b) -7x + 14y 8 = 0


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Solution.

6x + 3y = 8

3y = -6x + 8

y = -2x + 8/3

Hence,

Gradient = -2

y-intercept = 8/3

y = mx + c


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b) -7x + 14y 8 = 0

14y = 7x + 8

y = 1/2x + 4/7

Hence,

Gradient = 1/2

y-intercept = 4/7


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How to finding the equation of

a straight line


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Finding the equation of a

straight line

Straight line which is parallel to

the x-axis.In the figure, the line AB is

parallel to the x-axis


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Gradient of line AB,

m

y-intercept of line AB,

c =

By substituting the

values of m and c into

equation y=mx+c Y=

0

03

11

1

1


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The equation of the straight lineparallel to the x-axis with the

y-intercept, p

so,y=p


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What the equation of straight line PQ?


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Answer: y = 3


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Straight line which is parallel to

the y-axis

The gradient of a straight line parallel

to the y-axis is undefined as the

horizontal distance between any two

points on the line is always zero


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The straight line MN which is

parallel to the y-axis has constantx-coordinate is 4.

Thus , the equation of the straight

line MN isx = 4


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The equation of the straightline parallel to the y-axis with

the x- intercept, q

so, x = q


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What the equation of straight line RS?


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Answer: x = 3


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Straight line which passes through a given

point and has a specific gradient

The steps:

1) Substitute the given gradient m, the x-coordinate

and y-coordinate of the point (x,y) intoy = mx + c

2) Solve the equation in step 1 to fine the value ofc

3) Write the equation in the form ofy = mx+c, using

the value ofm and c found in step 2


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Example :

Find the equation of the

straight line passing through

the point A(-2,-5) which has agradient of 1/2


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Solution

Give that m =1/2, thereforey = 1/2x + cthe point A(-2,-5) satisfied the equation

above.

-5 =1/2(-2) + c

c = -5 + 1

= -4

Thus, the equation of the straight line is

y = 1/2x - 4


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Straight line which passes the two

points

Given two point, P ( ) and Q( ),

that lie on a straight line, the equation

of line PQ can be found by followingthe steps in bellow:

11, yx

22, yx


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Step 1: Find the gradient of line PQ,

Step 2: Find the value of c (y-intercept)by substituting the value of m

and the coordinates of one ofthe given point (P or Q) into

the equation y = mx + c

Step 3: Substitute the value of m andc into the equation y = mx + c

12

12

xx

yym


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Example:

Find the equation of the

straight line passingthrough the point

M(2,-1) and N(4,7)


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Solution 1

4

2

8

241)(7m

Gradient of line MN


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9)2(41

c

c

cmxy

Substitute m = 4, x = 2 and

y = -1 (coordinates of point M)

Into the equation.


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The equation of the line MN

IS y = 4x -9

Substitute m = 4 and c = -9

Into the equation

y = mx + c


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Find the equation of the straightline that passes through each pair

of points A(5,2) and B(3,10)


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THE GRADIENT

453

210

m

m


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What the value of c

22

)5(42

c

c

cmxy


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THE EQUATION IS ?

224 xy


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FINDING THE POINT OF

INTERSECTION

Complete the table of values and draw the

graphs of the straight line for the equation:

2 14 and 2 2 16y x y x

x 0 1 2 3 4

y 14

2 14y x

10 6 2 -2


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x 0 1 2 3 4

y 8

2 2 16y x

109 11 12


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The point of intersection of twostraight lines is the only point that

satisfies both equations. The point ofthe intersection of two lines can beobtained by:

a) drawing the graphs of two straightline

b) solving simultaneous equation


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CONCLUSION


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After verifiedm is a gradient, andc is

the y-interceptcan form the equation of

the straight line.y = mx+c

If the equation of the straight line in the

formax + by + c, transform it intoy = mx+c,then determinethe gradient

andy-intercept


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the equation of the straight lineparallel

tothex-axiswith they-intercept,p isy = p

the equation of the straight lineparallel

tothey-axiswith thex-intercept, q is

x = q

The equation of the straight line which passes


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The equation of the straight line which passesthrough a give point, (x,y) and has a specificgradient,m.

subtitute m, and (x,y) intoy = mx + cand find the value ofc

by that, form the equation

The equation of the straight line which passes

through two points.find the m

then subtitute m and (x,y) into the

y = mx + c to find cby that, form the equation


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To find the point of intersection of two

straight lines:

a) drawing the graphs of two straight line

b) solving simultaneous equation

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The Straight Line - Slide