Block 1

Intersection of Lines

What is to be learned?

• How to find where straight lines meet(point of intersection)

3x + 2y = 172x – 5y = 5

3x + 2y = 17 X5→ 15x + 10y = 852x – 5y = 5 X2→ 4x – 10y = 10

Add 19x = 95 x = 5

Consider two lines

simultaneous equations

15 + 2y = 17y = 1

POI (5 , 1)

(the Elimination Method)

Lines usually written:

y = 5x + 1y = 3x + 7

Could still use elimination!

→ y – 5x = 1→ y – 3x = 7

Lines usually written:

y = 5x + 1y = 3x + 7

So 5x + 1 2x + 1 = 7

2x = 6 x = 3

Point of intersection (3 , 16)

Must be the same

y = 5(3) +1 = 16

Known as

y=y tactic= 3x + 7

£y £y

Using y = y

Find the points of intersection, for these pairs of lines:

1. y = 5x – 3 and y = 2x + 122. y = 7x + 8 and y = x +203. y = -2x + 8 and y = 2x + 124. y = 5x + 3 and y = 24 – 2x

(5,22)(2,22)

(-1,10)

(3,18)

What is to be learned?

• How to find where straight lines meet(point of intersection)

y = 2x5x + 3y = 22

Handy!

y = 2x5x + 3y = 22

Replace y with 2x

5x + 3 = 225x + 6x = 22

11x = 22 x = 2 x = 2 y = 2(2)

= 4Intersection (2 , 4)

(2x)y

Handy!

substitution

y = 4x + 33x + 2y = 17

substitution

y = 4x + 33x + 2y = 17

Replace y with 4x + 3

3x + 2 = 173x + 8x + 6 = 17

11x + 6 = 17 11x = 11 x = 1 y = 4 (1) +3

= 7Intersection (1 , 7)

(4x+3)y

substitution

Using Substitution

Find intersections of these lines:1. x = 8 and

2y + 4x = 422. y = 8 and

3y = 2x + 43. y = 2x and

2y + 3x = 28

Using Substitution

Find intersections of these lines:1. x = 8 and

2y + 4x = 42 (8 , 5)2. y = 8 and

3y = 2x + 43. y = 2x and

2y + 3x = 28

Using Substitution

Find intersections of these lines:1. x = 8 and

2y + 4x = 42 (8 , 5)2. y = 8 and

3y = 2x + 4 (10 , 8)3. y = 2x and

2y + 3x = 28

Using Substitution

Find intersections of these lines:1. x = 8 and

2y + 4x = 42 (8 , 5)2. y = 8 and

3y = 2x + 4 (10 , 8)3. y = 2x and

2y + 3x = 28 (4 , 8)

Points of Intersection

Can use• Simultaneous Equations• y = y• Substitution

Depends on way lines are given

Ex 1 y = 5x – 3y = 3x + 5

Ex 1 y = 5x – 3y = 3x + 5

y = y5x – 3 = 3x + 5 2x = 8

x = 4 y = 5(4) – 3 = 17Intersection (4 , 17)

Ex 2 3x + 2y = 152x + 5y = 21

Simultaneous Equations

Ex 3 2x + 3y = 13 y = 2x – 1

Substitute y= 2x – 1 into*2x + 3(2x – 1) = 132x + 6x – 3 = 13 8x = 16 x = 2 y = 2 (2) – 1 = 3(from second equation)Intersection (2 , 3)

*

. You can often rearrange the equations to use your favourite tactic