Objective

LINEAR EQUATIONS

SOLVING SIMULTANEOUS EQUATIONS

BY ELIMINATION AND SUBSTITUTION

BACKGROUND KNOWLEDGE

SUBSTITUTION IN AN EXPRESSION

Evaluate the following expressions given that

x = 2, y = 3,

a = 1, b = -12.

a) 3x =3 x 2 = 6

b) 20 + 2a - b2

= 20 +2 x 1 - (-2)2

= 20 + 2 – 4

= 18

c) 6y - 2x - 2b = 6(3) - 2(2) - 2(-2)

= 18 - 4 - (-4)

= 18 - 4 + 4

= 18

Lowest common multiple by listing multiples.

(a) Find the L.C.M of the following pairs of numbers.

(i) 2 and 3

Multiples of 2 are: 2, 4, 6, 8, 10, 12 ----------

Multiples of 3 are: 3, 6, 9, 12, 15, 18 ------

Common multiples are: 6, 12 -------

The least common multiples (L.C.M) is 6

(ii) 5 and 7.

Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45 ------

Multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56 --------

Common multiples is 35, ------

Least common multiples (LCM)is 35.

Linear equations with one unknown.

Definition

A mathematical sentence with the symbol = (equals to) is called an equation.

Such a statement expresses the equality of things.

1 + 8 = 11 is an equation.

2a + 5 = 7 is an equation with one unknown.

Illustration

Since an equation states the equality of two things, it may be compared to a pair of scales. The content of the two scale pairs balance each other

Draw A SCALE

Solving linear equations with one unknown.

Solve the equation

15x - 3 + 12x – 8

Add 3 on both sides.

15x - 3 + 3 = 12x - 8 + 3

15x = 12x – 5

Subtract 12x on both sides

15x - 12x = 12x - 12x – 5

3x = -5

Divide 3 on both sides

3x = -5

3 3

x = -5/3

(b)

Solve the equation

t + 7 = 27 - 4t

Add 4t on both sides

t + 4t + 7 = 27 - 4t + 4t

Subtract 7 on both sides.

5t = 20

Divide by 5 both sides

5t = 20

5 5

t = 4

5 - x = x - 4 + 12

3 2

Multiply both sides by 6

(LCM of 3 and 2)

6(5 - x) = 6 (x - 4 + 12)

3 2

2(5 - x) = 3 (x - 4) + 6 x 12

8 - 2x = 3x - 12 + 72

9 - 2x = 3x + 60

Subtract 3x on both sides

-2x - 3x = 3x - 3x + 50

-5x = 50

Divide both sides by -5

-5x = 50

-5 -5

x = -10