Cambridge International AS and A Level Mathematics Pure Mathematics 1

The intersection of two lines

P1^

2

EXAMPLE 2.12 Find the co-ordinates of the vertices of the triangle whose sides have the
equations x + y = 4, 2x − y = 8 and x + 2 y = −1.

SOLUTION

A sketch will be helpful, so first find where each line crosses the axes.

^1 x^ +^ y^ = 4 crosses the axes at (0, 4) and (4, 0).

^2   2 x^ −^ y^ = 8 crosses the axes at (0, −8) and (4, 0).

^3 x^ +^2 y^ =^ −1 crosses the axes at ()^0 ,−^12 and (−1, 0).

Since two lines pass through the point (4, 0) this is clearly one of the vertices. It

has been labelled A on figure 2.24.

Point B is found by solving ^2 and ^3 simultaneously:

^2 × 2:^4 x−^2 y^ =^16

^3 :^ x+^2 y^ =^ −^1

Add 5 x = 15 so x = 3.

Substituting x = 3 in ^2 gives y = −2, so B is the point (3, −2).

Point C is found by solving ^1 and ^3 simultaneously:

^1 :^ x^ +^ y^ =^4

^3 :^ x^ +^2 y^ =^ −^1

Subtract −y= 5 so y= −5.

Substituting y = –5 in ^1 gives x = 9, so C is the point (9, −5).

2 x ± y 

x  2y ±

x  y 

±



±  x

y

2 A

B

C

±± 2

Figure 2.24