# An automobile company uses three types of steel S1, S2 and S3 for producing three types of cars C1, C2 and C3.

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An automobile company uses three types of steel S1, S2 and S3 for producing three types of cars C1, C2 and C3

Steel requirements (in tons) for each type of cars are given below :

 Cars Steel C1 C2 C3 S1S2S3 213 312 421

Using Cramer’s rule,

find the number of cars of each type which can be produced using 29, 13 and 16 tonnes of steel of three types respectively.

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Given : -

Steel requirement for each car is given

Let, Number of cars produced by steel type C1, C2 and C3 be x, y and z respectively.

Now,

We can arrange this model in linear equation system

Thus,

We have

2x + 3y + 4z = 29

x + y + 2z = 13

3x + 2y + z = 16

Here,

Solving determinant,

expanding along 3rd column

⇒ D = 1[30 – 25]

⇒ D = 5

⇒ D = 5

Again,

Solve D1 formed by replacing 1st column by B matrices

Here,

Solving determinant,

expanding along 3rd column

⇒ D1 = 1[( – 35)( – 3) – ( – 5)( – 19)]

⇒ D1 = 1[105 – 95]

⇒ D1 = 10

Again,

Solve D2 formed by replacing 2nd column by B matrices

Here,

Solving determinant,

expanding along 3rd column

⇒ D2 = 1[190 – 175]

⇒ D2 = 15

And,

Solve D3 formed by replacing 3rd column by B matrices

Here,

Solving determinant,

expanding along 1st column

⇒ D3 = – 1[ – 23 – ( – 1)3]

⇒ D3 = 20

Thus by Cramer’s Rule, we have

Thus,

Number of cars produced by type C1, C2 and C3 are 2, 3 and 4 respectively.

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