Question

Draw the graph of the pair of linear equations x − y + 2 = 0 and 4x − y − 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.

Answer 1

For x − y + 2 = 0   

When x = 1,

1− y + 2 = 0

−y + 3 = 0

y = 3

When x = 2,

2 − y + 2 = 0

4 − y = 0

y = 4

When x = 3,

3 − y + 2 = 0

5 − y = 0

y = 5

x	1	2	3
y	3	4	5

For 4x − y − 4 = 0  

When x = 1,

4(1) − y − 4 = 0

4 − y − 4 = 0

0 − y = 0

y = 0

When x = 2,

4(2) − y − 4 = 0

8 − y − 4 = 0

4 − y = 0

y = − 4

When x = 3,

4(0) − y − 4 = 0

0 − y − 4 = 0

y = − 4

x	1	2	0
y	0	4	−4

The triangle is formed by points A(2, 4), B(−2, 0) and C(1, 0).

Since both B and C lie on the x-axis, the base is the distance between their x-coordinates:

Base = |X_C − X_B|

= |1 − (−2)|

= 3 units

∴ The height is the perpendicular distance from vertex A to the x-axis, which is its y-coordinate:

Height = 4 units

∴ Area = `1/2 xx "Base" xx "Height"`

= `1/2 xx 3 xx 4`

= `1/2 xx 12`

= 6 sq. units