6 6.1equations of straight lines 6.2general form of equations of straight lines chapter summary case...

6

6.1 Equations of Straight Lines

6.2 General Form of Equations of Straight Lines

Chapter Summary

Case Study

Equations of Straight Lines

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In the figure, we observe that the straight line passes through the points (0, 0), (1, 1), (2, 2) and (3, 3).

Case Study

Hm ... Can you tell me?

Do you know how to represent this straight line by an equation?

For all these points, the x-coordinates and the corresponding y-coordinates are the same.

Therefore, we can represent the straight line by the equation y x.

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A. IntroductionA. Introduction

6.1 Equations of Straight Lines6.1 Equations of Straight Lines

In junior forms, we learnt that the graph of a linear equation in two unknowns is a straight line.

In the figure, each straight line passes through different points.

The line y x passes through the point (2, 2) while the line x 2y 5 passes through the point (1, 2).

For every points on a straight line, their x- and y-coordinates must satisfy the equation of the straight line:

For the line x 2y 5 and the point (1, 2),L.H.S. 1 2(2)

5 R.H.S.

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B. Special Forms of Straight LinesB. Special Forms of Straight Lines


1. Horizontal Lines (lines that are parallel to the x-axis)

In the figure, the straight line L passes through A(–4, 4), B(–1, 4), C(0, 4), D(2, 4) and E(5, 4).

The y-coordinates of all these points are equal to 4.

∴ The equation of L is y 4.

In general, the equation of a horizontal line with y-intercept k is given by:

The y-intercept is 4.

y k

Notes:The equation of the x-axis is y 0.

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B. Special Forms of Straight LinesB. Special Forms of Straight Lines


2. Vertical Lines (lines that are perpendicular to the x-axis)

In the figure, the straight line L passes through A(–2, 5), B(–2, 3), C(–2, 0), D(–2, –1) andE(–2, –3).

The x-coordinates of all these points are equal to –2.

∴ The equation of L is x –2.

In general, the equation of a horizontal line with x-intercept k is given by:

The x-intercept is –2.

x k

Notes:The equation of the y-axis is x 0.

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C. Finding Equations of Straight Lines under C. Finding Equations of Straight Lines under Different ConditionsDifferent Conditions


1. Given the slope of the straight line and the coordinates of a point on it

In the figure, we can draw infinity many lines passing through point A.

Given a fixed point on a straight line and its slope, it is sufficient to determine that straight line.

y – y1 m(x – x1)

However, only 1 line can be drawn which passes through point A and is parallel to the Line L.

Consider a point A(x1, y1) lying on a straight line with slope m. P(x, y) is any point on the straight line.

Slope of PA Slope of the line m

xx

yy

1

1

This is known as the point-slope form of the equation of the straight line.

numbers

variables

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Example 6.1T

Solution:(a) The equation of the straight line:

Consider a straight line L with slope which passes through A(–1, 0).

(a) Find the equation of the straight line.(b) Determine whether P(1, 4) and Q(3, 2) lie on the line L.

2

1

)]1([2

10 xy

2

1

2

1 xyThe equation of a straight line can be written in the form of y as the subject.

P. 8



Example 6.1T

Solution:

Consider a straight line L with slope which passes through A(–1, 0).

(a) Find the equation of the straight line.(b) Determine whether P(1, 4) and Q(3, 2) lie on the line L.

2

1

For P(1, 4):L.H.S. 4

R.H.S. 2

1)1(

2

1 1

∵ L.H.S. R.H.S.

For Q(3, 2):L.H.S. 2

R.H.S. 2

1)3(

2

1 2

∵ L.H.S. R.H.S.

∴ P(1, 4) does not lie on the line L.

∴ Q(3, 2) lies on the line L.

(b) Substitute each of the values of into the equation :2

1

2

1 xy

P. 9



Example 6.2T

Solution:

If a line passes through (0, 6) and the inclination is 60, find the equation of the line.

∵ Inclination 60 ∴ Slope of the line tan 60 3

The equation of the straight line:

)0(36 xy63 xy

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2. Given the slope and the y-intercept of the straight line

In the figure, line L with slope m passes through a point on the y-axis with the y-intercept c.

y mx c

∴ Line L passes through the point (0, c).


y – y1 m(x – x1) y – c m(x – 0) y – c mx

This is known as the slope-intercept form of the equation of the straight line.

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Example 6.3T

Solution:The equation of the straight line:

22

3 xy

Find the equation of the straight line with y-intercept –2 and slope .2

3

Since the y-intercept is –2 and the slope of the straight line is given, we can apply the formula directly to find the equation of the straight line.

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3. Given the coordinates of any two points on the straight line

In the figure, A(x1, y1) and B(x2, y2) are two points on the straight line L.

∴ y – y1 m(x – x1)

In this case, we can first find the slope of L by using the formula

.slope12

12

xx

yy

)( 112

121 xx

xx

yyyy

Remarks:The above equation looks rather complicated.It is a good practice to identify/find the slope of a straight line first, then follow the point-slope form to find the equation of the straight line.

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Example 6.4T

Solution:

Find the equation of straight line with x-intercept 2 and y-intercept 6.

The straight line passes through two points (2, 0) and (0, 6).

20

06

∴ Slope of the line

–3


)2(30 xy

Using the point (2, 0): Using the point (0, 6):

63 xy

)0(36 xy

63 xy

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Example 6.5T(a) Find the equation of straight line passing through (0, 0) and (–3, –9).(b) If P(3, k) lies on the above straight line, find the value of k.



Solution:(a) Slope of the line

03

09

3 The equation of the straight line:

)0(30 xyxy 3

(b) Substitute x 3, y k into the equation of straight line, we havek 3(3)

9

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A. General Form of Equations of Straight LinesA. General Form of Equations of Straight Lines

6.2 General Form of Equations of6.2 General Form of Equations of Straight Lines Straight Lines

From the previous section, we learnt that the equations of straight lines can be written in different ways.

In general, the equation of a straight line can be expressed in the form

Ax By C 0, where A, B and C are constants.

This is called the general from of the equation of a straight line.

Notes:1. A, B and C can be any real numbers, but A and B cannot be both zero.

2. The right hand side of the general form of the equation is always zero.

For example: Rewrite into the general form:)3(2

12 xy

3)2(2 xy342 xy

072 yx

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B. Features of Equations of Straight LinesB. Features of Equations of Straight Lines


The general form of the equation of a straight line Ax By C 0 can be rewritten as:

Compare with the slope-intercept form y mx c, we have:

B

Cx

B

Ay

CAxBy

Slope B

A y-intercept B

C

If we substitute y 0 into the general form of the equation a straight line Ax By C 0,

0)0( CBAx

x-intercept A

C

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Remarks:1. If B 0 but A 0, then the equation becomes Ax C 0.

2. If A 0 but B 0, then the equation becomes By C 0.

Thus, x represents a vertical line. A

C

Thus, y represents a horizontal line. B

C

The straight line does not have any y-intercept and the slope is undefined.

The straight line has y-intercept and the slope 0. B

C

3. Ax By C 0 can be rewritten as , where B 0.B

Cx

B

Ay

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Example 6.6T

Solution:

Consider a straight line L: 6x 3y 9.(a) Find the slope and the y-intercept of the straight line.(b) Find the equation of the straight line with the same y-intercept as L

and its slope is 5.

(a) Rewrite the equation of the straight line in the general form, we have 6x 3y 9 0, i.e., 2x y 3 0.



Slope 1

2

y-intercept 1

3

3

2

(b) The equation of the line: y 5x 3 (or 5x y 3 0)

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C. Points of Intersection of Two LinesC. Points of Intersection of Two Lines


In junior forms, we learnt the algebraic method of solving simultaneous equations to find the point of intersection of two non-parallel straight lines on the coordinate plane:

Consider L1: 2x y 1 0 and L2: y x 5.

)2( ......5

)1( ......012

xy

yx

Substituting (2) into (1), we have2x (x 5) 1 0 3x 6 x 2

Substituting x 2 into (2), we have

y 2 5 3

)2( ......05

)1( ......012

yx

yx

(1) (2),(2x y 1) (x y 5) 0 3x 6 0 x 2

Substituting x 2 into (2), we have

2 y 5 0 y 3 ∴ The point of intersection of L1 and L2 is (2, 3).

Method of substitution Method of elimination

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Remarks:Given two straight lines:

L1: y m1x c1 L2: y m2x c2

Either one of the following cases will happen:

1. One point of intersection

different slopes, i.e., m1 m2

2. No intersection

same slope, i.e., m1 m2

different y-intercepts, i.e., c1 c2

3. Infinitely many points of intersection

same slope, i.e., m1 m2

same y-intercept, i.e., c1 c2

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Example 6.7T

Solution:

Consider the following straight lines:L1: 4x 3y 7 0 and L2: 3x 2y 18 0 (a) Find the point of intersection of P.(b) Find the equation of the straight line passing through P and the origin.

(a)

)2( ......01823

)1( ......0734

yx

yx

(1) 3: 12x 9y 21 0 ...... (3)(2) 4: 12x 8y 72 0 ...... (4)

(3) (4): 17y 51 0 y 3

Substituting y 3 into (2), we have 3x 2(3) 18 0 x 4

∴ The point of intersection is (4, 3).

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Example 6.7T

Solution:

Consider the following straight lines:L1: 4x 3y 7 0 and L2: 3x 2y 18 0 (a) Find the point of intersection of P.(b) Find the equation of the straight line passing through P and the origin.

(b) Slope of the line 04

03

4

3

Equation of the line:

xy4

3

or 3x 4y 0The straight line passes through (0, 0). Thus the y-intercept is 0.

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D. Parallel Lines and Perpendicular LinesD. Parallel Lines and Perpendicular Lines


Consider two straight lines L1 and L2 with slopes m1 and m2 respectively. 1. If L1 // L2 , then m1 m2 .

2. If L1 L2 , then m1 m2 1.

Conversely, if m1 m2 , then L1 // L2 .

Conversely, if m1 m2 1, then L1 L2 .

P. 24


Consider a straight line L1: 6x 4y 3 0. If another line L2 passes through (1, 1) and is parallel to L1, find the equation of L2 .


Slope of L1 4

6

2

3

The equation of L2 :

)1(2

31 xy

)1(3)1(2 xy

3322 xy

0123 yx

Example 6.8T

Solution: Alternative Solution:Rewrite the equation of L1:

6x 4y 3 0

4

3

2

3364

xy

xy

Let the equation of L2 be .cxy 2

3 ∵ L1 // L2

Substitute (1, 1) into the equation,

we obtain c .2

1

2

1

2

3 xy∴

P. 25


Consider the following lines L1: 2x y 7 0 and L2: x 4y 1 0.(a) Find the point of intersection of the two lines.(b) Find the equation of the straight line passing through the point of

intersection of L1 and L2 which is perpendicular to L2.


Example 6.9T

Solution:

(a)

)2( ......014

)1( ......072

yx

yx

(1): 2x y 7 0(2) 2: 2x 8y 2 0 ...... (3)

(1) (3): 9y 9 0 y 1

Substituting y 1 into (2), we have x 4(1) 1 0 x 3

∴ The point of intersection is (3, 1).

P. 26


Consider the following lines L1: 2x y 7 0 and L2: x 4y 1 0.(a) Find the point of intersection of the two lines.(b) Find the equation of the straight line passing through the point of

intersection of L1 and L2 which is perpendicular to L2.


Example 6.9T

Solution:

(b) Slope of L2 4

1

Since the required line L2 , we have

slope of the required line 1

4

1

slope of the required line 4

Equation of the required line:

)3(4)1( xy1241 xy

0134 yx

The required line passes through the point of intersection of L1 and L2, which is (3, 1).

P. 27

6.1 Equations of Straight Lines

Chapter Summary

1. Special Forms of Straight Lines(a) The equation of a horizontal line with y-intercept k is y k. (b) The equation of a vertical line with x-intercept k is x k.

2. Finding equations of straight lines under different conditions (a) Given the slope of the straight line and the coordinates of

point on it:y – y1 m(x – x1)

(b) Given the slope and the y-intercept of the straight line:y mx c

(c) Given the coordinates of any two points (x1, y1) and (x2, y2)on the straight line:

Slope of the line

Then use the point-slope form to find the equation of thestraight line.

12

12

xx

yy

P. 28


Chapter Summary

1. The general form of the equation of a straight line is Ax By C 0.2. From the general form of the equation of a straight line, for A, B 0,

we have:

(a) slope B

A

(b) x-intercept A

C

(c) y-intercept B

C

3. By solving simultaneous equations, we can find the point of intersection of two non-parallel straight lines in the coordinate plane.

P. 29


Chapter Summary

4. (a) Two straight lines will intersect at one point only if they havedifferent slopes.

(b) Two straight lines will not intersect if they have the same slopebut different y-intercepts.

(c) Two straight lines have infinitely many points of intersection ifthey have the same slope and y-intercept.

6 6.1equations of straight lines 6.2general form of equations of straight lines chapter summary case...

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