Question

The area of the triangle formed by the lines 2x + 3y = 12, x – y – 1 = 0 and x = 0 (as shown in Fig.) , is

A. 7 sq. units

B. 7.5 sq. units

C. 6.5 sq. units

D. 6 sq. units

Answer 1

Given

Equation 1: 2x + 3y = 12

Equation 2: x – y = 1

Equation 3: x = 0

To calculate the Area at first we solve the Equation 1 & 2 Simultaneously by method of substitution.

We substitute the value of x from Equation 2 in Equation 1 to get the value of y

Equation 2:

x – y = 1

⇒ x = y + 1

Equation 1:

2x + 3y = 12

Substituting the value from equation 2 we get

2(y + 1) + 3y = 12

⇒ 2y + 2 + 3y = 12

⇒ 5y = 10

⇒ y = 2

Putting the value in Equation 1 we get

x = 2 + 1

⇒ x = 3

So both this lines passes through ( 3 , 2) Let this Coordinate name be P₁

Equation 3 is the equation for y axis

Equation 1 meets Y axis at (0 ,4) which is calculated by substituting x = 0 in Equation 1. Let this Coordinate name be P₂

Equation 2 meets Y axis at (0 , – 1) which is calculated by substituting x = 0 in Equation 2. Let this Coordinate name be P₃

So Area of the triangle = \(\frac{1}{2}\) x | x₁ x (y₂ – y₃) + x₂ x (y₃ – y₁) + x₃ x (y₁ – y₂) |

Where x₁ ,y₁ are the coordinates of P₁

x₂, y₂ are the coordinates of P₂

x₃ ,y₃ are the coordinates of P₃

⇒ Area of the Given Triangle = \(\frac{1}{2}\) x | 3 x (4 +1) + 0 x ( – 1 – 2) + 0 x (2– 4) |

⇒ Area of the Given Triangle = \(\frac{1}{2}\) x | 3 \(\frac{1}{2}\) 5 |

⇒ Area = \(\frac{15}{2}\)

⇒ Area = 7.5 Sq.

Units Area of the triangle is 7.5 sq. Units