i. Theorem (without proof) Cramerâ€™s Rule, ii. Non Homogeneous linear equations upto three variables

**Cramerâ€™s Rule**

Gabriel Cramer, a Swiss mathematician born in the city Geneva in
31 July 1704. He edited the works of the two elder Bernoulliâ€™s, and wrote on
the physical cause of the spheriodal shape of the planets and the motion of
their apsides (1730), and on Newtonâ€™s treatment of cubic curves (1746).

In 1750 he published Cramerâ€™s Rule, giving a general formula for
the solution of certain linear system of n equations in n unknowns having a
unique solution in terms of determinants. Advantages of Cramerâ€™s rule is that
we can find the value of *x*, *y* or *z* without knowing any of
the other values of *x*, *y* or *z*. Cramerâ€™s rule is applicable
only when D â‰ 0 ( is the determinant
value of the coefficient matrix) for unique solution.

Consider,

If AX = B is a system of n linear equations in â€˜nâ€™ unknowns such
that det (A) â‰ 0, then the system has a unique solution.

This solution is,

where *A _{j}* is the matrix obtained by replacing the
entries in the

(a) Consider the system of two linear equations with two unknowns.

The solution of unknown by Cramerâ€™s rule is

(b) Consider the system of three linear equations with three
unknowns *a*_{1}

*x *+* b*_{1}* y *+* c*_{1}* z *=* d*_{1}

*a*_{2}* x *+* b*_{2}* y *+* c *_{2}* z *=* d*_{2}

*a*_{3}* x *+* b*_{3}* y *+* c *_{3}* z *=* d*_{3}

**Example 1.19**

Solve the equations 2*x* + 3*y* = 7, 3*x* + 5*y*
= 9 by Cramerâ€™s rule.

*Solution:*

The equations are

2*x* +
3*y* = 7

3*x* +
5*y* = 9

Here

âˆ´ we can apply Cramerâ€™s
Rule

âˆ´ By Cramerâ€™s rule

âˆ´ Solution is *x *=* *8,* y *= âˆ’3

**Example 1.20**

The following table represents the number of shares of two
companies *A* and *B* during the month of January and February and it
also gives the amount in rupees invested by Ravi during these two months for
the purchase of shares of two companies. Find the the price per share of *A*
and *B* purchased during both the months

*Solution:*

Let the price of one share of *A* be *x*

Let the price of one share of *B* be *y*

âˆ´ By given data, we get
the following equations

10*x* +
5*y* = 125

9*x* +
12 *y* = 150

âˆ´ By Cramerâ€™s rule

The price of the share *A* is â‚¹10 and the price of the share *B* is â‚¹5.

**Example 1.21**

The total cost of 11 pencils and 3 erasers is â‚¹ 64 and the total
cost of 8 pencils and 3 erasers is â‚¹49. Find the cost of each pencil and each
eraser by Cramerâ€™s rule.

*Solution:*

Let â€˜*x*â€™ be the cost of a pencil

Let â€˜*y*â€™ be the cost of an eraser

âˆ´ By given data, we get
the following equations

11x + 3y =64

8x+3y=49

âˆ´ The cost of a pencil is â‚¹ 5 and the cost of an eraser is â‚¹ 3.

**Example 1.22**

Solve by Cramerâ€™s rule *x* + *y* +
*z* = 4, 2*x* âˆ’ *y* + 3*z* = 1, 3*x* + 2 *y* âˆ’ *z* = 1

*Solution:*

âˆ´ We can apply Cramerâ€™s
Rule and the system is consistent and it has unique solution.

The solution is ( x , y,
z) = ( âˆ’1,3, 2)

**Example 1.23**

The price of 3 Business Mathematics books, 2 Accountancy books and
one Commerce book is â‚¹840. The price of 2 Business Mathematics books, one
Accountancy book and one Commerce book is â‚¹570. The price of one Business
Mathematics book, one Accountancy book and 2 Commerce books is â‚¹630. Find the
cost of each book by using Cramerâ€™s rule.

*Solution:*

Let â€˜*x*â€™ be the cost of a Business Mathematics book

Let â€˜*y*â€™ be the cost of a Accountancy book.

Let â€˜*z*â€™ be the cost of a Commerce book.

âˆ´ 3*x* + 2 *y* + *z* = 840

2*x* +
*y* + *z* = 570

*x *+* y *+* *2*z *=* *630

âˆ´ The cost of a Business
Mathematics book is â‚¹120,

the cost of a Accountancy
book is â‚¹150 and

the cost of a Commerce book is â‚¹180.

**Example 1.24**

An automobile company uses three types of Steel *S*_{1},
*S* _{2} and *S*_{3} for providing three different
types of Cars *C*_{1}, *C*_{2} and *C*_{3}.
Steel requirement *R* (in tonnes) for each type of car and total available
steel of all the three types are summarized in the following table.

Determine the number of Cars of each type which can be produced by
Cramerâ€™s rule.

*Solution:*

Let â€˜*x*â€™ be the number of cars of type *C*_{1}

Let â€˜*y*â€™ be the number of cars of type *C*_{2}

Let â€˜*z*â€™ be the number of cars of type *C*_{3}

3*x* +
2 *y* + 4*z* = 28

*x *+* y *+* *2*z *=* *13

2*x* +
2 *y* + *z* = 14

âˆ´ The number of cars of
each type which can be produced are 2, 3 and 4.

Tags : Theorem with Solved Example Problems , 12th Business Maths and Statistics : Chapter 1 : Applications of Matrices and Determinants

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12th Business Maths and Statistics : Chapter 1 : Applications of Matrices and Determinants : Cramerâ€™s Rule | Theorem with Solved Example Problems

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