MATHEMATICS FORM ONE TOPIC 4- COORDINATE GEOMETRY(2)
UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU:0787237719
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MATHEMATICS FORM ONE TOPIC 4- COORDINATE GEOMETRY(2)
(d) (2,6), (5,3)
Let (x1, y1) = (2,6)
(x2, y2) = (5, 3)
It has a negative gradient
(e) (1,6), (3, -1)
Let (x1, y1) be (1, 6)
(x2, y2) be (3, -1)
The gradient is negative
(f) (3, 6), (-2, -1)
Let (x1, y1) be (3, 6)
(x2, y2) be (-2, -1)
m =
=
=
The gradient is positive
(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
The gradient is positive
(i) (2,10), (2,0)
Let (x1, y1) be (2, 10)
(x2, y2) be (2, 0)
M
=
M is undefined
The gradient is undefined
(j) (–), , 2)
The gradient is positive
(k) (-2,1), (4,3)
let (x1,y1) be (-2,1)
(x2,y2) be (4,3)
The gradient is positive
(i ) (-4,4), (-3,-3)
let (x1,y1) be (-4,-4)
(x2,y2) be (-3,-3)
M
m= -7
The gradient is negative
(m) (0,0), (-3,4)
let (x1,y1) be (0,0)
(x2,y2) be (-3,4)
M
The gradient is negative
(n) (99,6), (119,1)
let (x1,y1) be (99,0)
(x2,y2) be (119,1)
M
The gradient is negative
(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
The gradient is zero
EXERCISE 10.3
- 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.
b) (0,5) with gradient -2
(x, y), ( 0, 5) m = -2
-2(0 – x) = 1(5-y)
0+2x = 5 –y
y= 5 -2x
∴ y= -2x + 5
- c) (1,-3) with gradient -3
(x, y) , (1, -3) m = – 3
-3 – y = 3 – 3x
-y = – 3 + 3x + 3
-y = 3x
y= -3x
- 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
= +
- 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
- 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
- g) ( 6, 2)and y intercept -2
(6, 2), (0, -2)
Gradient = 2/3
Arbitrary
(x,y), (6,2)
3(2-y) = (6-x)
6-3y = 12 – 2x
-3y = 12 – 6-2x
-3y = 6- 2x
- 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
(0, – 2) gradient 1
Arbitrary (x, y)
y – 0 = y + 2
x = y+2
y = – 2 + x
(ii) y-intercept 7,gradient 3/4
(0,7), (x,y) , gradient 3/4
iii) y-intercept -16, gradient 4
(0,-16), (x,y) m = 4
4x-0 = y +16
-y =-4x +16
y = 4x -16
y = 4x – 16
- 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
Gradient = 2
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
Gradient = 24
y-intercept = (0,0)
vi) 6x =5-2y
x=0
Gradient = -3
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
