Chapter 3 Linear Equations

Module PMR

CHAPTER 3 : LINEAR EQUATIONS 1.

Solve linear equation = Finding the value of the unknown which satisfies the equation.

2.

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

3.

A linear equation in one unknown has only one root.

4.

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. There are 4 different forms of linear equation as follow: Equation x+a=b

Solution x=b-a

x-a=b

x=b+a

ax = b

x

x b a

b a

x=axb

Solving Linear Equations in One Unknown involving combined operations of +, -, x,  . 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 right side. 3. Solve the equation using combined operations. 4. Check your solution by substituting the value into the original equation.

Examples: 1. Given that 2y + 11 = -5, calculate the value of y. Solutions:

Linear Equations

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Module PMR

2 y  11  5 2 y  5  11 2 y  16 16 y 2 y  8 2. If k  1 

k 5 ,find the value of k. 3

3. Find the value of q which satisfies the equation 3(q  4)  q  2

Solution:

Solution: k 5 3 3(k  1)  k  5 3k  3  k  5 3k  k  5  3 2k  8 8 k 2 k 4 k 1 

3(q  4)  q  2 3q  12  q  2 3q  q  2  12 2q  14 14 q 2 q7

4. If 7 - (x + 1) = -4x, then x = ? Solution: 7  ( x  1)  4 x 7  x  1  4 x  x  4 x  7  1 3 x  6 6 x 3 x  2 Common Errors No Errors m  2  7 1. m  7  2 m  5 x5  3 2. x  3  5 x8

Linear Equations

Correct Steps m  2  7 m  7  2 m  9 x5  3 x  35 x  2

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Module PMR

3.

2y  8 8 y 2 y4

2y  8 y  8 2 y  16 1 p 8 4

4.

p  8

1 p 8 4 p  8 4 p  32

1 4

p2

5.

1.

2x  3  7 2x  7  3 2 x  10 10 x 2 x5 Exercise a) x  2  10

2x  3  7 2x  7  3 2x  4 x  4 2 x8 Example x  5  11 x  11  5 x6

b) x  4  2

c) 4  x  7

d) 10  x  3

e) 6  x  3

f) 3  7  x

Linear Equations

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Module PMR

2.

x 3  5 x  53 x8

a)

x 8  6

b)

x  10  3

c) 7  x  2

d) 1  x  9

e) 3  x  7

f) 5  9  x

3.

5  6 x x  65 x 1

a) 6  7  x b) 5  4  x

c) 3   x  2

d) 1  x  10

e)  x  8  3

f) 5  x  7

Linear Equations

35

Module PMR

4.

2 x  10 10 x 2 x  5

a) 3 x  18 b) 5 x  25 c) 8  2x d) 16  4x e)  x  10 f) 3 x 

5.

x 2 5 x  25 x  10

a)

1 2

x  5 2

b) 

x 2 7

c)

x 1  2 3

d)

5x  10 3

2 1 e)  x  5 4

f) 

Linear Equations

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x 3  6 2

Module PMR

6.

2x 1  5 2x  5 1 2x  6 6 x 2 x3

a) 3 x  4  5

b) 6  4 x  2

c)

1 x 3  4 2

d) 3 

2 x  7 5

e) 4 

x 1 2

f) 9  3 

7.

3x  1  2 x  4 3 x  2 x  4  1 x  5

3 x 2

a) 3 x  4 x  7

b) 4 x  9  3  2 x

c) 5  x  11  3x

d) Linear Equations

37

x 2 x4 3

Module PMR

e) 2 x =

5x − 3 =x 4

f)

8.

3( x − 2) = 2 x + 5 3x − 6 = 2 x + 5 3x − 2 x = 5 + 6 x = 11

4 + 9x 5

a)

4( x − 3) − 6 = x

b) x − 3( x + 1) = 9

c) x + 4 = 3 − 2( x − 5)

d)

1 (2 x − 3) = −5 + 2 x 3

e)

x −5 x = 3 6

f)

PMR past year questions 2004 1). Solve each of the following equations. Linear Equations

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2 x − 5 3x + 4 = 3 2

Module PMR

a) k = −14 − k 3 b) f + (6 − 4 f ) = −31 2

[3 marks] 2005 2). Solve each of the following equations. a) 2n  3n  4 b) 2k 

3  7k 5

[3 marks] 2006 3). Solve each of the following equations. a)

12 3 n

b) 2(k  1)  k  3

[3 marks] 2007 4). Solve each of the following equations. a) x  10  4 Linear Equations

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Module PMR

b)

5x  4 x 3

[3 marks] 2008 5). Solve each of the following equations. a) p  5  11 b) x  1 

x3 2

[3 marks]

CHAPTER 3 :LINEAR EQUATIONS ANSWERS 1.

a) x = -12 b) x = -2

Linear Equations

2. 40

a) x = 14 b) x = 7

Module PMR

c) x = 3 d) x = -13 e) x = -9 f) x = -10 a) x = 1 b) x = 9 c) x = -1 d) x = -9 e) x = 11 f) x = 2

3.

5.

4.

a) x = -10 b) x = -14 2 c) x = 3 d) x = 6 5 e) x =  8 f) x = 9 a) x = 7 b) x = -3 c) x = 8 d) x = -9 e) x = 4 f) x = 3

7.

c) x = 9 d) x = -8 e) x = 10 f) x = 4 a) x = 6 b) x = -5 c) x = 4 d) x = -4 e) x = 10 1 f) x = 6 a) x = -3 b) x = 2 c) x = 14 d) x = -25 e) x = 6 f) x = -4

6.

8.

a) b) c) d) e)

x=6 x = -6 x=3 x=3 x = 10 22 f)  5

PMR past year questions 2004.

2005.

1). a). k = -7

2). a). n = 4

b). f = 8

b). k =

3 17

2006.

2007.

3). a). n = 4

4). a). x = -6

b). k = 5 2008. 5). a). p = -16

Linear Equations

b). x = 2 b). x = 5

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