Question

Find the area of the triangular region whose vertices are the points of intersection of the graphs 2x + y = 5, y = x − 4 and y = 5.

Answer 1

Given

2x + y = 5

y = x − 4

y = 5y 

Step 1: Find points of intersection

Intersection of 2x + y = 5 and y = 5

Substitute y = 5 into the first equation

2x + 5 = 5

2x = 0

x = 0

Point (0, 5)

Intersection of y = x − 4 and y = 5

Substitute y = 5

5 = x − 4

x = 5 + 4

x = 9

Point (9, 5)

Intersection of 2x + y = 5 and y = x − 4

Substitute y = x − 4 into 2x + y = 5

2x + (x − 4) = 5

3x − 4 = 5

3x = 9

x = `9/3`

x = 3

then y = 3 − 4 = −1

Point (3, −1)

Step 2: Use the vertices to find the area

We now have the triangle with vertices:

A(0, 5)

B(9, 5)

C(3, −1)

Area = `1/2 |x_1(y_2 − y3) + x_2(y_3 − y_1) + x_3(y_1 − y_2)|`   ... [use the area of triangle method]

= `1/2|0(5 + 1) + 9(−1 − 5) + 3(5 − 5)|`

= `1/2|0 − 54 + 0|`

= `1/2 xx 54`

Area = 27 square units