NB: Kyk elke dag die video vir verduideliking van die dag ... · Oefening/ Exercise 1 p 47 no a) 2,...
Transcript of NB: Kyk elke dag die video vir verduideliking van die dag ... · Oefening/ Exercise 1 p 47 no a) 2,...
Graad 10 Hoofstuk 4 - Vergelykings en Ongelykhede Grade 10 Chapter 4 - Equations ______________________________________________________________________________________
Dag 22 / Day 22 NB: Kyk elke dag die video vir verduideliking van die dag se werk
en die memo van elke oefening.
NB: Watch the video each day for explanation of the work of the
day and the memo of the exercise.
Lineêre Vergelykings Linear Equations
Voorbeelde Examples
Los op vir x: Solve for x:
a) 2 3 2 3( 3)x x+ = − +
2 3 2 3 9
2 3 3 7
5 10
2
x x
x x
x
x
+ = − −
+ = − −
= −
= −
b) 5(1 3 ) 2(10 ) 10x x x− − − = −
5 15 20 2 10
15 13 10
3 15
5
x x x
x x
x
x
− − + = −
− − = −
− =
= −
Voorbeelde
Reël: Om ‘n breuk in ‘n vergelyking weg te kry moet met die KGV aan beide kante gemaal word.
Examples
Rule: To solve a fraction in an equation, multiply with the LCD to get rid of the fraction
1. Los op vir y:
10 1 4(7 1)
2 5
y y+ +=
KGV = 10:
5(10 1) 4(2)(7 1)
50 5 56 8
6 3
1
2
y y
y y
y
y
+ = +
+ = +
− =
= −
2. Los op vir x:
3 1 6 LCD = 4
2 4
2(3) (1) 6(4)
6 24
18
18
xx x
x
x
x
x
− =
− =
− =
− =
= −
Oefening/ Exercise 1 p 47 no a) 2, 3, 6, b) 2, 4, 6
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 2
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 3
Dag 23 / Day 23 Kwadratiese vergelykings Quadratic equations
Om die vergelyking op te los: Maak die RK = 0
Faktoriseer die LK
Stel die faktore = 0
To solve the equation: Put RHS = 0
Factorise LHS
Let the factors = 0
Voorbeelde Examples
Los op vir x: Solve for x:
a) 2 2 8 0x x− − =
( 4)( 2) 0
4 or 2
x x
x x
− + =
= = −
b) (2 1)( 2) 2x x− + = −
2
2
2 3 2 2
2 3 0
(2 3) 0
30 or
2
x x
x x
x x
x x
+ − = −
+ =
+ =
= = −
Oefening / Exercise 2 p 49 no a) 2, 3, 6, b) 4, 13, 16, 22,24
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 4
Dag 24 / Day 24 Breuke vergelykings: Fraction equations:
Voorbeelde: Examples:
1. 3𝑥−4
𝑥2−3𝑥−4+
4
4−𝑥=
5
2𝑥+2
3𝑥−4
(𝑥−4)(𝑥+1)−
4
(𝑥−4)=
5
2(𝑥+1) 𝑥 ≠ 4; 𝑥 ≠ −1
KGV = 2(𝑥 − 4)(𝑥 + 1)
3𝑥−4
(𝑥−4)(𝑥+1)×
2(𝑥−4)(𝑥+1)
1−
4
(𝑥−4)×
2(𝑥−4)(𝑥+1)
1=
5
2(𝑥+1)×
2(𝑥−4)(𝑥+1)
1
2(3𝑥 − 4) − 8(𝑥 + 1) = 5(𝑥 − 4)
6𝑥 − 8 − 8𝑥 − 8 = 5𝑥 − 20
6𝑥 − 8𝑥 − 5𝑥 = −20 + 8 + 8
−7𝑥 = −4
𝑥 =4
7
Onthou om altyd jou antwoord te toets in breuk vergelykings
Remember to always test your answer in fraction equations
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 5
2) 2 2
3 5 4
3 2 4 1
x
x x x x
−− =
+ + − − −
3 5 4
;( 2)( 1) ( 2)( 2) 1
x
x x x x x
− −− =
+ + + − + 2; 2; 1x x x − − KGV = (x + 2)(x – 2)(x + 1)
( 3)( 2) 5( 1) 4( 2)( 2)x x x x x− − − + = − + −
2 2
2 2
2
2
5 6 5 5 4( 4)
10 1 4 16
5 10 15 0
5( 2 3) 0
5( 3)( 1) 0
3 or 1 but 1
x x x x
x x x
x x
x x
x x
x x x
− + − − = − −
− + = − +
− − =
− − =
− + =
= = − −
Slegs x = 3.
Oefening / Exercise 3 p 51 no a) 2, 6, 7, 8, b) 2, 5, 7
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 6
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 7
Dag 25 / Day 25 Gelyktydige Vergelykings Simultaneous Equations
Oplossing van gelyktydige vergelyking deur
substitusie:
Solution of simultaneous equations by
substitution:
1. Los op vir x en y:
Solve for x and y:
x + 2y = 5 en x – y = - 1
2 5 ........(1)
1 ........(2)
5 2 ........(3)
x y
x y
x y
+ =
− = −
= −
Stel (3) in (2)
5 2 1
5 3 1
3 6
2
5 2(2)
5 4
1
y y
y
y
y
x
x
x
− − = −
− = −
− = −
=
= −
= −
=
(1; 2) is die oplossing. Is the solution
Los op vir x en y deur eliminasie:
Solve for x and y by elimination:
1. Los op vir x en y:
Solve for x and y:
a) x + 2y = 5 en x – y = - 1
2 5 ........(1)
1 ........(2)
(1) - (2): 3 6
2
x y
x y
y
y
+ =
− = −
=
=
x + 2(2) = 5
x = 1
(1; 2) is die oplossing. Is the solution.
b) 3x + 2y = 2 en 5x – 2y = - 18
3 2 2 ........(1)
5 2 18 ........(2)
(1) + (2): 8 = 16
2
x y
x y
x
x
+ =
− = −
−
= −
3( 2) 2 2
6 2 2
2 8
4
y
y
y
y
− + =
− + =
=
=
(- 2 ; 4) is die oplossing. Is the solution
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 8
Oefening / Exercise 4 p 53 no b), d), e), f) .
Dag 26 / Day 26
Oefening / Exercise 5 p 55 no a) 2, 3, 4, b) 2, 4, 5
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 9
Dag 27 / Day 27 Letter Vergelykings Letter Equations
1. Los op vir x: Solve for x: a)
mx p q
mx q p
q px
m
+ =
= −
−=
b) A = 2 2( )R r −
Maak r die onderwerp van die formule.
Make r the subject of the formula:
Oplossing:
2 2
2 2
2 2
2 2
2 2
2
( )R r A
AR r
Ar R
Ar R
Ar R
Ar R
− =
− =
− = −
= − −
= −
= −
Radius kan slegs positief wees.
Radius can only be positive.
c) Los op vir x:
Solve for x:
2( ) ( )
2 2
2 2
(2 ) 2
2
2
(2 )
2
x a b x a
x a bx ab
x bx a ba
x b a ba
a bax
b
a bx
b
+ = −
+ = −
− = − −
− = − −
− −=
−
− +=
−
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 10
Oefening / Exercise 6 p 57 no a) 2, b) 2, 3, c) 3, d), e) 2
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 11
Dag 28 / Day 28
Lineêre Ongelykhede Linear Inequalities
Voorbeelde: Examples:
Los op vir x: Solve for x:
1. 2 3
3( 1)2
xx
− + LCD = 2
2 3 6( 1)
2 3 6 6
4 9
9
4
x x
x x
x
x
− +
− +
−
−
Getallelyn:
Number line:
2. 4 3 1 5x− −
3 3 6
1 2
x
x
−
−
Getallelyn:
Number line:
1. 2 3
3( 1)2
xx
− + KGV = 2
2 3 6( 1)
2 3 6 6
4 9
9
4
x x
x x
x
x
− +
− +
−
−
Getallelyn:
Number line:
2. 4 3 1 5x− −
3 3 6
1 2
x
x
−
−
Getallelyn:
Number line:
Oefening / Exercise 7 p 60 no a) 2, 4, b) 5, c) 2, 4, 6
-4 -3 -2 -1 9
4−
-1 2
-4 -3 -2 -1 9
4−
-1 2
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 12
Hersiening Oefening p 70 – Als op jou eie (opsioneel)
Revision Exercise p 70 - Everything on your own (optional)
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 13
Dag 29 / Day 29
GRAAD 10 GREY KOLLEGE
WISKUNDE Werkskaart 1 TOTAAL: 34
GRADE 10 GREY COLLEGE
MATHS WORKSHEET 1 TOTAL: 34
VRAAG 1 QUESTION 1
Los op vir x: Solve for x:
1.1. ( )( ) 357 =−− xx (4)
1.2. 33
14
−−
x (4)
1.3. −4 < 2 − 2𝑥 < 6 (3)
1.4. (x – 2)(2x + 1) = x + 4 (4)
1.5. 3( 3) 2( 1)x x− = + (4)
1.6. 5( 8) (8 3 ) 5 3( 8)x x x x+ − − = − + (6)
1.7.
5 3 413
4 9
x x− −− =
(5)
1.8. 2 36x = (4)
[34]
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 14
Dag 30 / Day 30
GRAAD 10 GREY KOLLEGE
WISKUNDE Werkskaart 2 TOTAAL: 15
GRADE 10 GREY COLLEGE
MATHS WORKSHEET 2 TOTAL: 15
Vraag 1 QUESTION 1
Los op vir x: Solve for x:
Los op vir x en y gelyktydig:
Solve for x and y simultaneously:
Graad 10 Hoofstuk 4 Vergelykings
Grey Kollege 15
Dag 31 / Day 31
GRAAD 10 GREY KOLLEGE
WISKUNDE Werkskaart 3 TOTAAL: 38
GRADE 10 GREY COLLEGE
MATHS WORKSHEET 3 TOTAL: 38
VRAAG 2 / QUESTION 2
Los op vir x : / Solve for x:
2.1. 53𝑥 = 25𝑥+1
(3)
2.2. 3.2𝑥+1 − 192 = 0 (4)
2.3. (3𝑥 − 2)(𝑥 + 1) = 0 (4)
2.4. 2𝑥2 = 6𝑥 (4)
2.5. 2(𝑥 − 2)(𝑥 + 4) = 32 (5)
2.6. 2𝑥
3−
5
3=
3𝑥
2−
5
2 (6)
2.7. 3(𝑥 − 2) − (4𝑥 + 1) < −3 (4) [30]
VRAAG 3 / QUESTION 3
3.1. Los op vir x en y gelyktydig: / Solve x and y simultaneously:
6𝑥 − 𝑦 − 4 = 0 9𝑥 + 1 = −2𝑦 (5)
3.2. Maak a die onderwerp van die volgende: / Make a the subject of the following:
𝑥2 = 𝑎2 − 𝑦2 (3) [8]