chapter 3 i linear equations enhance

7/29/2019 Chapter 3 I Linear Equations ENHANCE

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3.1 Solutions of Linear Equations in One Unknown

Solve linear equations = Finding the value of the unknown which satisfies the

equation.

The solution of the equation is also known as the root of the equation.

A linear equation in one unknown has only one root.

To determine whether a given value is a solution of an equation, substitute the value

into the equation. If the sum of the left hand side (LHS) = sum of right hand side

(RHS), then the given value is a solution.

3.2 Solving Linear Equations in One Unknown through the operations +, , and

There are 4 different forms of linear equation as follows :

Type Equation Operation Solution

I x + a = b x = b a

II x a = b + x = b + a

III ax = b x =

a

b

IV0, = ab

a

x x = a b

Examples :

3.3 Solving Linear Equations in One Unknown involving combined operations of +, , and

Linear equations in the form of ax + b = c

Steps :

1. Work on the bracket first, if there is any.

2. Group the terms with the unknown on the left hand side of the equation while the

numbers on the equation using combined operations.3. Solve the equation using combined operations.

4. Check your solution by substituting the value into the original equation.

x + 2 = 6 x = 6 2 = 4

x 3 = 4 x = 4 + 3 + 7

4x = 12 x =4

12= 3

-5x = 30 x =5

30

= -6

2

x= 6 x = 6 2 = 12

8

x= 3 x = 3 (-8) = -24

CHAPTER 3

LINEAR EQUATIONS


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Common Mistake

1.

WRONG RIGHTa. x 2 = 7

x = 7 2

x = 5

a. x 2 = 7

x = 7 + 2

x = 9

b. y + 6 = 1

y = 1 + 6

y = 7

b. y + 6 = 1

y = 1 6

y = -5

c. 2m = 10

m = 10 x 2

m = 20

c. 2m = 10

m = 10 2

m = 5

d.4

1n = 16

n = 16 x4

1

n = 4

d.4

1n = 16

n = 16 4

1

n = 64

2.

WRONG RIGHTa.

2 - (w + 5) = 4

2 - w + 5 = 4

2 + 5 - 4 = w

3 = w

a.

2 (w + 5) = 4

2 w 5 = 4

2 5 4 = w

-7 = w

b.

-3(w 2) = 7

-3w - 6 = 7

-3w = 7 + 6w

=3

13

b.

-3 (w 2 ) = 7

-3w + 6 = 7

-3w = 7 6w

=3

1

c.

5 2 2 (w 1 ) = 6

3 (w 1 ) = 6

3w 3 = 6

3w = 6 + 3

c.

5 2 (w 1 ) = 6

5 2w + 2 = 6

5 + 2 6 = 2w

1 = 2w


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w = 3

9

w = 3

2

1 = w

Addition / Subtraction

EXAMPLE EXERCISE 1

1. m 3 = 5

m = 5 + 3

m = 8

1. m 8 = 6

2. m + 4 = 7

m = 7 4

m = 3

2. m + 3 = 9

3. -3 + m = -2

m = -2 + 3

m = 1

3. -5 + m = 3

4. 3 + m = -2

m = -2 3

m = -5

4. 2 + m = -3

5. 7 = m 3

7 + 3 = m

10 = m

m = 10

5. 3 = m 7

6. -7 = m 3

-7 + 3 = m

-4 = m

m = -4

6. 3 = m + 6

7. 6 = -3 + m

6 + 3 = m

9 = m

m = 9

7. 3 = -6 + m

8. -6 = 3 + m

-6 3 = m

-9 = m

m = -9

8. -3 = 7 + m


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Addition / Subtraction

EXAMPLES EXERCISE 2

1. 5 = 6 m

m = 6 5

m = 1

1. 6 = 7 m

2. -5 = 6 - m

m = 6 + 5

m = 11

2. -6 = 5 m

3. 5 = -m + 6

m = 6 5

m = 1

3. 6 = - m + 8

4. -5 = -m + 6

m = 6 + 5

m = 11

4. -6 = - m + 5

5. 5 m = 1

5 1 = m

4 = m

5. 1 m = 5

6. 5 m = -2

5 + 2 = m

7 = m

6. 2 m = -5

7. m 2

1=

4

1

m =2

1

4

1+

7. m +3

2=

3

1


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m =4

3

EXERCISE 3 EXERCISE 4

1. x + 1 = 10

1. w +4

3=

2

1

2. 2 + x = -2

2. w 4

1=

3

1

3. x 4 = 3

3. w 1 =3

2

4. u 9 = -11

4. w +

5

1= 2

5. - u + 2 = 3

5. 2 + w =2

1

6. 4 u = 7 6. 2 = w 31

7. 4 = 3 u7. 1 4w =

3

1 5w


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Division

EXAMPLES EXERCISE 1

1. 2m = 8

m =2

8

m = 4

1. 3m = 8

2. 2m = -8

m =2

8

m = - 4

2. 3m = 5

3. -2m = 8

m =2

8

m = -4

3. -4m = 7

4. -2m = -8

m = 2

8

m = 4

4. -4m = -9

5.2

1m = 4

m = 4 2

1

m = 2

5.3

1m = 5

6.

2

1m = - 4

m = -4 2

1

m = -8

6.

3

1m = -5

7.3

2 m = 6

m = 6 3

2

7.5

2 m = 10


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m = -9

8.3

2 m = -6

m = 6 3

2

m = 9

8.5

2 m = - 10

Multiplication

EXAMPLES EXERCISE 1

1.5

m= 2

m = 52

m = 10

1.2

m= 3

2.5

m= -2

m = - 52m = -10

2.7

m= -3

3.5

m = 2

m = )5(2

m = -10

3.7

m = 3

4.5m = -2

m = - 52

m = 10

4.7m = -3

5.3

m=

2

1

m = 32

1

m = 2

3

5.7

m=

2

3

6.3

m =

2

1

m = )3(2

1

m =2

3

6.6

m =

2

3


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7. m = 6

m = )1(6

m = -6

7. m = 10

8. m = -6

m = )1(6

m = 6

8. m = -10

Solve each of the following equation:EXAMPLES EXERCISE 5

1. 4p = 12

p =4

12

p = 3

1. -6p = 1

2. w31 = 6

w =3

16

w = 18

2.34

9= w

3. 1 - 2w = 5

1 - 5 = 2w

2

4= w

-2 = w

3. 1 - w =3

2

4.2

w + 1 = 4

2

w = 4 1

2

w = 3

w = 23 = 6

4. 5 = 2 +3

w

5. 2n + 1 = n + 6

2n n = 6 1

n = 5

5. 2n + 1 = 2 n

6. 5m 2 = 2 m 1

5m 2m = -1 + 2

3m = 1

6. 5m + 2 = 4 2m


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m =3

1

7. 1 2m = 4 m 3

1 + 3 = 4m + 2m

4 = 6m

6

4= m

3

2= m

7. 7m 3 = 2 m 8

EXAMPLES EXERCISE 6

1.2

3+m= 6

m + 3 = 26m + 3 = 12

m = 12 3

m = 9

1.2

6+w= 9

2.2

3+m= -6

m + 3 = - 26

m + 3 = -12

m = -12 3

m = -15

2.6

3w= -2

3.7

12 m= 5

2m 1 = 752m 1 = 35

2m = 35 + 1

m =2

36= 18

3.7

12 +w= -3

4. 2 =4

9 m

42 = 9 mm = 9 8

m = 1

4.3

1=

6

2 w

5.31 =

912 +m

93

1 = 2m + 1

3 1 = 2m

2 = 2m

2

2= m

5.32 =

62 w


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6.3

1+m= m

m + 1 = 3mm + 1 = 3m

1 = 3m m

1 = 2m

2

1= m

6. 2w =4

12 +w

EXAMPLES EXERCISE 7

1. 4 ( m 2 ) = 1

4m 8 = 1

4m = 1 + 8

4m = 9

m =49

1. 3 ( k 2 ) = 4

2. 4 = 2 ( m + 1)

4 = 2m + 2

4 2 = 2m2 = 2m

2

2= m

1 = m

2. 24 = 3 ( k + 1)

3. 3 ( 1 + m ) - 2 = 43 + 3m - 2 = 4

3m = 4 - 3 + 2

3m = 3

3m = 3

m =3

3= 1

3. 2 ( 1 + k ) - 4 = 2

4. 2 - 3(2 + m) = 4

2 - 6 - 3m = 4

2 - 6 - 4 = 3m-8 = 3m

3

8 = m

4. 4 - 2(2k + 3) = 2

5. 2 - (2 + m) = 4

2 - 2 - m = 4-m = 4

m = -4

5. 3 - (k + 4) = -2

6. 3 - (2 - m) = 4

3 - 2 + m = 4

6. 3 - (4 - 2k) = 2


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m = 4 - 3 + 2

m = 3

7. m + 5 = 2 - 3(2m - 1)

m + 5 = 2 - 6m + 3

m + 6m = 2 + 3 - 5

7m = 0

m =70 = 0

7. 2k + 2 = 7 - 3(k + 5)

EXERCISE 8 EXERCISE 9

1.3

10m= m

1. 2k + 5 = 1 ( k + 5)

2.3

2w = -3

2. 1 2(1 k ) = k + 2

3. 4 = 1 3

1w

3. 3 = 4k 3( k + 2)

4.w

2= 8

4. 7 ( 3k + 5) = 3k

5.w

3 2 = 1

5. 2 4k = -3 ( k 2)


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6.w

w 13 =

3

4 6. k 4 = 2 ( 4 k )

Questions based on examination format

1. Given that m

3

2+ 2 = 6, then m =

A. 6 B. 8 C. 10 D. 12

2. Given that4

27 +m= 3, then m =

A. -4 B. -2 C. 2 D. 4

3. Given that 7

5

34

+m= 4, then m =

A. -6 B. -4 C. 3 D. 6

4. Given that 15

2m = 9, then m =

A. 4 B. 20 C. 25 D. 50

5. Given that3

1(1 6q) + 9 = 4(q 3), therefore q =

A.9

32B.

3

16C. 18 D. 32

6. Given that5

2p 3 = 5, then p =

A. 20 B. 10 C. 5 D. 1


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7. Given that m4

1+ 1 = 5, then m =

A. 1 B. 16 C. 19 D. 24

8. Given that 6 2(m 3) = m, then m =

A. 0 B. 1 C. 3 D. 4

9. Given that4

53 k+= 2k + 3, then k =

A. 0 B. -2 C. -3 D. -5

10. Given that 4 -3

2(6k 1) = 3k, then k =

A.

3

1B.

3

2C.

3

5D.

3

7

11. Given that3

1(m + 3) = 4 - 2(m + 5), therefore m =

A. -3 B. -2 C. -1 D. 0

12. Given that 2(3q + 7) - 3(q 2) = -1, therefore q =

A. -2 B. -7 C. 4 D. 5

13. Given that 3(1 2p) = 2(4p 9), therefore p =

A.3

2B.

4

3C.

5

6D.

2

3

14. Given that 2 -3

m= m - 2, then m =

A. 6 B. 3 C. 0 D. -3

15. Given that 2 -3

y= 4, then y =

A. 10 B. 6 C. -6 D. -10

16. Given that x8

1 2 = 6, then x =

A. 1 B. 32 C. 64 D. 81

17. Solve the equation7

2k+ 3 = k 2.


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A.7

1B. 3

2

1C. 3

4

3D. 7

18. Solve the equation 9m + 5(m 2) = 60.

A.7

43 B. 5 C. 12

2

1D. 17

2

1

19. Solve the equation3

2m+ 2 = 2.

A. -12 B. -6 C. 0 D. 24

20. Given that 2 -3

2p = - 6, then p =

A. 6 B. 12 C. -6 D. -12

Past Year SPM Questions

November 2003

1. Given that p3

1+ 1 = 4, thenp =

A. 1 B. 9 C. 11 D. 15

July 2004

2. Given that 7(k 3) = 1 4k, then k=

A. 2 B. 7 C.

3

4D.

11

4

November 2004

3. Given that 3k 4 = 5(2 k) then k=

A. -7 B. -3 C.2

7D.

4

7

July 2005

4. Given that

4

5

2

=h

, then h =

A.5

8B.

8

5C.

2

5D.

2

3

November 2005

5. Given that 10 3(2 w) = 9w + 2, calculate the value ofw.

A.6

1B.

4

1C.

3

1D.

5

2


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July 2006

6. Given that 9 3(k 2) = 0 , then k=

A. 1 B. 2 C. 5 D. 8

November 2006

7. Given that ),3(242

5kk = then k =

A.12

17B.

10

17C.

6

17D.

4

17

Jun 2007

8. Given that 9 3(k 8) = 3 k , find the value ofk

A. -9 B. -1 C. 15 D. 18

November 2007

9. Given that ,125

1=

+x

xfind the value of x =

A.9

2B.

11

4C.

9

5D.

3

2

Jun 2008

10. Given that ,2)23(

5

1=+ nn calculate the value of n =

A. 6 B. 2 C. 2 D. -6

November 2008

11. Given that p + 2 =4

)21(3 pfind the value ofp =

A.2

1 B.

5

1 C.

10

1D.

6

1


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ANSWERS


17/18

Chapter 3 Linear Equation

Addition/Subtraction

1 14 2 6 3 8

4 5 5 10 6 3

7 9 8 10

Exercise 21 1 2 11 3 2

4 11 5 -4 6 77 - 1/3

Exercise 3

1 9 2 -4 3 74 -2 5 -1 6 -3

7 -19

Exercise 41 - 2 7/12 3 1 2/3

4 1 4/5 5 -1 6 -37 -1

Exercise 5

1.

6

1

2. -12 3.

3

21

4. 9 5.

3

1

6.

7

2

7. -1

Exercise 6

1. 12 2. -9 3. -11

4. 0 5. 3 6.

6

1

Exercise 7

1.

3

10

2. 7 3. 2

4. -1 5. 1 6.

2

3

7. -2

Exercise 8

1. -5 2.

2

9

3. -9

4.

4

1 5. 1 6.

5

3

Exercise 9

1. -3 2. 3 3. 94.

3

1 5. -4 6. 4

Division

Exercise 1

1 8/3 2 5/3 3 -7/4

4 9/4 5 15 6 -15

7 -25 8 25

MultiplicationExercise 1

1 6 2 -21 3 -21

4 21 5 21/2 6 97 -10 8 10

SPM Format

Exercise 1

1 -1/6 2 -12 3 1 2/3

4 9 5 1/3 6 2/77 -1

Exercise 2

1 12 2 -9 3 -11

4 0 5 3 6 1/6

Exercise 3

1 10/3 2 7 3 2

4 -1 5 1 6 3/2

7 -2

Linear Equations 1


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Exercise 4

Exercise 5

SPM Format

1 A 2 B 3 C

4 C 5 A 6 A

7 B 8 D 9 C10 B 11 A 12 B

13 D 14 B 15 C

16 C 17 D 18 B

19 B 20 B

SPM PAST YEAR QUESTIONS

1 B 2 D 3 A

4 C 5 C 6 C

7 A 8 C 9 D

10 A 11 A

Li E ti

1 -5 2 -9/2 3 -9

4 5 1 6 3/5

1 -3 2 3 3 94 1/3 5 -4 6 4

2

chapter 3 i linear equations enhance

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