MATHEMATICS FORM ONE TOPIC 4- COORDINATE GEOMETRY(2)

MATHEMATICS FORM ONE TOPIC 4- COORDINATE GEOMETRY(2)

UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU:0787237719

MATHEMATICS FORM ONE TOPIC 4- COORDINATE GEOMETRY(2)

(d) (2,6), (5,3)

Let (x1, y1) = (2,6)

(x2, y2) = (5, 3)

(e) (1,6), (3, -1)

Let (x1, y1) be (1, 6)

(x2, y2) be (3, -1)

(f) (3, 6), (-2, -1)

Let (x1, y1) be (3, 6)

(x2, y2) be (-2, -1)

m =

=
=

(g) (0,2), (6,2)

Let (x1, y1) be (0, 2)

(x2, y2) be (6, 2)

M =

==

= 0

M =0 The gradient is zero

(h) (2,3), (-1,-3)

Let (x1, y1) be (2, 3)

(x2, y2) be (-1, -3)

m = 2

(i) (2,10), (2,0)

Let (x1, y1) be (2, 10)

(x2, y2) be (2, 0)

M

=

M is undefined

(j) (–), , 2)

(k) (-2,1), (4,3)

let (x1,y1) be (-2,1)
(x2,y2) be (4,3)

(i ) (-4,4), (-3,-3)

let (x1,y1) be (-4,-4)
(x2,y2) be (-3,-3)

M

m= -7

(m) (0,0), (-3,4)

let (x1,y1) be (0,0)
(x2,y2) be (-3,4)

M

(n) (99,6), (119,1)

let (x1,y1) be (99,0)
(x2,y2) be (119,1)
M

(o) (0.64,-1.62), (1.36,-0.62)

let (x1,y1) be (0.64,-1.62)
(x2,y2) be (1.36,-0.62)

M

m=0

EXERCISE 10.3

1. a) (2,1) with gradient 2.

(x, y), (2, 1) m= 2

2(2-x) = 1(1-y)

4-2x = 1-y

4 -2x =1-y

y=1-4+2x

y= -3+2x

∴y = 2x – 3.

(x, y), ( 0, 5) m = -2

-2(0 – x) = 1(5-y)

0+2x = 5 –y

y= 5 -2x

∴ y= -2x + 5

1. c) (1,-3) with gradient -3

(x, y) , (1, -3) m = – 3

-3 – y = 3 – 3x

-y = – 3 + 3x + 3
-y = 3x

y= -3x

1. d) (-2, -4) with gradient 3/2

(x, y) (-2 , -4) m = 3/2
= =

3(-2 – x) = 2(-4 – y)

-6 – 3x = -8 – 2y

2y = 6 + 3x – 8

=  +

1. e) (0, 0 ) with gradient -3

(x,y) (0,0) m = -3

=  =

-3(0 – x) = 1(0 – y)

0 +3x = 0 – y

y = 0 – 3x

y = -3x

1. f) (-3 , -3) and y- intercept 1/2

Solution
(-3, -3)(x, y) y – intercept =1/2

Y = mx + c

(-3, -3),(x, y)

substitute (-3,-3) to y = mx +c

1. g) ( 6, 2)and y intercept -2

(6, 2), (0, -2)

Arbitrary

(x,y), (6,2)

3(2-y) = (6-x)
6-3y = 12 – 2x
-3y = 12 – 6-2x
-3y = 6- 2x

1. h) (-1 , -1 )and y – intercept – 1/3

(-1, -1) and (0, -1/3)

Arbitrary point = (x,y)
(x,y), (-1,-1)

2(-1-x) = 3(-1-y)
-2-2x=-3-3y
3y=-3+2+2x
3y=-1+2x

i) (1,2) and y-intercept = 2

(1,2), (0,2)

0 = 2-y
y = 2

j) (5,5) and y-intercept 0

(5,5) (0,0)

y = 5-5 + x
y=x

(2) (i) y – intercept -2 , gradient 1

Arbitrary (x, y)

y – 0 = y + 2

x = y+2

y = – 2 + x

(0,-16), (x,y) m = 4

4x-0 = y +16
-y =-4x +16
y = 4x -16
y = 4x – 16

1. iv) y-intercept 2, gradient is -10

3 (i) 7x + 4y = 11

Alternatively

7x +7y = 11
4y = -7x +11

ii) 14x + 3y = 12

iii) 2x=5 +y

x=0
2x = 5 + y, y=0-5, y =-5
y = 2x -5
y =2x-5
y- intercept = (0,2)

iv) 4x +5y= 40

x=0

5y= -4x + 40

y = 24x
y = 24 (0)
y = 0
y = -8x X -3 = 24x
y = 24x + 0

y-intercept = (0,0)

vi) 6x =5-2y

x=0

4. x-intercept , y=0

5x + 6(0) – 60 = 0
5x – 60 = 0 + 60

x= 12
X- intercept = (12,0)

Y- intercept, x=0
5(0) + 6y – 60 = 0
5x – 60 = 60

y = 10
y-intercept = (0,10)

Area = 60 square units

EXERCISE 10.4
1. (i) 2x + y = 5

4x – y = 7

x = 2
2x + y =5
2 (2) + y = 5
4 + y = 5
y = 5-4
y = 1

x =2 and y = 1

(ii). 3x + y = 6
5x + y =6

x = 1
from 3x + y = 6
3 x 1 + y = 6
3 +y = 6
y = 6-3
y = 3

x = 1 and y = 3

(iii) 5x -2y = 16
x + 2 = 8

x = 4
Let find the value of y by taken equation one
5x -2y = 16
5 x 4 – 2xy =16
20 – 2y = 16
-2y = 16 – 20

y = 2
x = 4 and y = 2

(iv) 8x + 5y = 40
9x + 5y = 5

x+0 = 35
x = 35
let find the value of y by taken one equation

8 x 35 + 5y =40
280 + 5y = 40

y = -48
x = 35 and y = -48

(v). 7x -4y = 17
5x -4y = 11

Let find the value of y taken one equation 7x – 4y = 17

(vi) 0.7x – 0.5y = 2.5
0.7x – 03y = 2.9

0-0.2 = -0.5
0.2y = -10.4
0.2 = -10.4
y = 2

Let find the value of x by taken one equation

0.7x – 0.5y = 2.5
0.7x – 0.5(2) = 2.5
0.7x = 2.5 + 1

2. (i) 3x -2y = 5
2x +y = 8

3x – 2y = 5……………(i)
2x + y = 8…………….(ii)
y=8 – 2x ……………..(iii)

put eqn (iii) into (iv)
3x -2x (8 -2x) = 5
3x – 16 + 4x = 5
3x + 4x -16 =5
7x = 5 + 16

x=3
y= 8-2 x3
y= 8-6
y = 2
x =3 and y = 2

(ii) 5x + y =23
3x – 2y = 6

5x – y = 23…………(i)
3x -2y = 6………….(ii)
5x-y +y = 23 + y.
5x =23 + y
5x – 23 =y…………….(iii)

put eqn (iii) in (ii)

3x -2 (5x – 23) = 6

3x -10x + 46 = 6
-7x+ 46 = 6
7x = 6 – 46

Let find the value of y by taken one equation

(iii). x -3y=2

4x + 2y = 36

x -3y=2 ……………(i)
4x + 2y = 36 …………(ii)
x=2 + 3y ……………..(iii)

put eqn (iii) in (ii)

4 (2 + 3y) + 2y = 36
8 + 12y +2y = 36
8 + 14y = 36
14y = 36 -8

y = 2

Let find the value of x by taken one equation

x= 2 +3y
x = 2+3(2)
x = 2 +6
x = 8

y = 2 and x = 8

(iv) 7x – y = 14
8x – 2y = 16

7x – y =11……………(i)
8x – 2y = 16 …………(ii)
7x – 14 = y ………………(iii)

put eqn (iii) into eqn (ii)

8x – 2x (7x – 14) = 16
8x – 14 + 28 = 16
-6x + 28 = 16

x = 2

Let find the value of y by taken one equation

y = 7 x2 = 14
y = 14 -14 = 0
y = 0

x = 2 and y =o

(v) 7x + y = 14
8x – 2y =6

7x + y =14…………….(i)
8x – 2y = 6…………… (ii)
y = 14 – 7x ……………(iii)

put eqn (iii) in (ii)

8x – 2(14 – 7x) =6
8x – 28 + 14x =6
22x – 28 = 6

x = -90

Let find the value of y by taken one equation

x = -90 and y = 27
(i) 3y – x = 4
y + 2x = 6

x = 2
Let find the value of y by taken one equation

y +2x = 6
y + 2 x2 = 6
y +4 =6
y = 6-4
y =2

x =2andy = 2

(ii) 8m- n = 38
m – 3n = -1

n = 2

Let find the value of m by taken one equation
m -3 = -1
m-3 X 2 = -1
m-6 = -1
m = -1 + 6
m = 6-1
m = 5

n = 2 and m = 5

(iii) 5x – 2y = 10
-x + 3y = 24

MATHEMATICS FORM ONE TOPIC 4- COORDINATE GEOMETRY(2)

Let find the value of y by taken one equation

5 x 6 – 2y = 10

30 – 2y = 10

-2y = 10 -30

y = 10

x = 6 and y = 10

4. (i) x -y = -3
2x – y = -5

x-y = -3 ………………..(i)
2x – y = -5 …………….(ii)
x=-3 + y ……………….(iii)

put eqn (iii) into eqn (ii)
2(-3 + y) – y= -5
-6 + 2y – y = -5
y – 6 = -5
y = -5 + 6
y = 1

x = -3 + y
= -3 +1
x = -2

x= -2 and y = 1

(ii). x -2y = 6
x + 2y = 2

x – 2y = 6 ………………..(i)
x + 2y = 2 ……………….(ii)
x =6 + 2y ………………..(iii)

put eqn 3 into eqn 2

6 + 2y +2y = 2
6 + 4y = 2
4y = 2 -6

y = -1

x = 6 + 2x -1
x = 6 + -2
x= 4

x = 4 and y = -1

(iii) 3x – 4y = -11
2x + 3y = 16

6x -4y = -11 ………………(i)
6x + 3y = 16………………(ii)
6x = -11 + 4y …………….(iii)

put eqn (iii) into (ii)
-11 + 4y + 3y = 16
-11 + 7y = 16
7y = 16 +11

(iv) 2x – 3y = 32
3x – 4y = 30

6x – 9y =96…………..(i)
6x – 8y = 60…………..(ii)
6x = 96 + 9y ………….(iii)

put eqn (iii) into eqn (ii)

96 + 9y -8y =60
96 + y =60
y = 60 – 96
y= -36

6x = 96 +9x -36
6x = 96 – 324

x = -38 and y = -36

(v) 5a -5b = 7
2a – 4b = 2

10a-5b =7 ………….(i)
10a -4b = 2 ………….(ii)
10a = 7 + 5b ………….(iii)

put eqn (iii) into eqn (ii)i

7 +5b -4b= 2
7 + b =2
b =2-7
b = -5
10a = 7+5x -5
10a = 7 + -25
10a =7 – 25

5. (i) 10u + 3v – 4 = 0
6u + 2v – 2 = 0

10u + 3v = 4
6u + 2v = 2

let find the value of u by taken equation
10u + 3v =4
10u + 3 (-2) = 4
10u -6 = 4
10u = 4 +6
10u = 10

v = -2 and u = 1

(ii) x – y = 1

let find the value of x by taken one equation

(iii) 3x + 3y =15
2x + 5y = 14

Let find the value of x by taken one equation

3x + 3y =15

(iv) 7x -3y = 15
5x – 2y = 19

y = 58
Let find the value of x taken one equation
7x -3y =15
7x – 3 x 68 = 15
7x – 204 =15

(v). x + y =5
x – y =1

y = 2

Let find the value of x by taken one equation

x + y = 5
x + 2 = 5
x= 5-2
x = 3
x = 3 and y = 2

2x + 4y = 48………………… (ii)

Now you can take equation (i) and (ii) to solve the equation

3x-4y =12
2x+ 4y = 48

x = -9

Let find the value of y by taken one equation

3x – 4y =3

3(-9) -4y = 3

-27 – 4y = 3

-4y = 3 + 27

y = -7.3 and x = – 9

EXERCISE 10.5

1. Let the numbers be x and y

2y = 80

y = 40
let find the value of x
x + y = 109
x + 40 = 109
x = 109 – 40
x = 69
The two numbers are 69 and 40

2. let the first number be x
let the second number be y

x + 3y = 1 ……………(i)

Let find the value of y

x+3y =1

3. Let the girls number be x and the boys be y

x = 16
let find the value of x

x + y = 36
16 + y = 36
y = 20

The girls are 16 and boys are 20

4. let the length be x and the width be y

Let find the value of y

2x -3y = 1
2 x 2 -3y = 1
4 – 3y = 1

y = 1
The length =2cm and width = 1cm

5. Let the pencils be x and pen be y

x = 50

Let the value of x

4x + 5y = 600
4 x 50 + 5y = 600
200 +5y = 600
5y =600 -200

y = 80

The pencils are 50 and pens are 80

6. let paul,s money be x

let john’s money be y

y = 1500

Let find the value of x

x = 2200
Paul’s money = 2200 shs and John’s money = 1500 shs

7. let the price of the sheep be a
Let the price of goat be = b

2a = 100

a = 50
let find the value of b

3a + 4b = 290
3 x 50 + 4b = 290
150 + 4b = 290
4b = 290 – 150

b = 35

The price of sheep = a
The price of goat = b
They bought 1 goat at 35 shs and 1 sheep at 50shs