Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

Created by: Arshit Thakur

B Com : Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

The document Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev is a part of the B Com Course Business Mathematics and Statistics.
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SOLUTION OF LINEAR EQUATIONS  unique solution given by X = A-1B. Example 24 Solve using matrices the equations 2x- y = 3, 5x+y = 4. Solution :
The equations can be written in matrix form as

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev


∴ The unique solution is given by

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Example 25 Solve the equations 2x +8y +5z = 5, x +y +z = -2, x +2y -z = 2 by using matrix method.
Solution : The equations can be written in matrix form as
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Now

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Example 26 
A woman invested different amounts at 8%, 8 4 % and 9%, all at simple interest. Altogether she invested Rs. 40,000 and earns Rs. 3,455 per year. How much does she have invested at each rate if she has Rs. 4,000 more invested at 9% than at 8%? Solve by using matrices.

Solution :
Let x, y, z be the amounts in Rs. invested at 8%, 8 3 % and 4
9% respectively.
According to the problem,
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

The equations can be written in matrix form as 
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
∴ The unique solution is given by X = A-1B We now find A-1 .

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Now,

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Now,

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Hence the amounts invested at 8%, 8 3 /4% and 9% are 4 Rs. 11,000, Rs. 14,000 and Rs. 15,000 respectively.

Solution by Determinant method (Cramer’s rule) Let the equations be
a1 x + b1 y + c, z = d1 ,
a2 x + b2 y + c2 z = d2 ,
a3 x + b3 y + c3 z = d3 .


Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
 

When Δ ≠ 0, the unique solution is given by

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Example 27 Solve the equations x +2y +5z = 23, 3x +y +4z = 26, 6x +y +7z = 47 by determinant method.

Solution :
The equations are
x +2y +5z = 23
3x +y +4z = 26
6x +y +7z = 47

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

By Cramer's rule

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Example 28 Solve the equations 2x-3y-1 = 0, 5x +2y -12 = 0 by Cramer’s rule.
Solution :
The equations are 2x - 3y = 1, 5x +2y = 12

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

By Cramer's rule

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Example 29 A salesman hasthe following record ofsales during three months for three items A, B and C which have different rates of commission.
Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Find out the rates of commission on the items A, B and C. Solve by Cramer’s rule.

Solution :
Let x, y and z be the rates of commission in Rs. per unit for A, B and C items respectively.

According to the problem,
90x +100y +20z = 800
130x +50y +40z = 900
60x +100y +30z = 850

Dividing each of the equations by 10 throughout,
9x +10y + 2z = 80
13x + 5y + 4z = 90
6x + 10y + 3z = 85
 

Now,

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

By Cramer's rule

Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Hence the rates of commission for A, B and C are Rs. 2, Rs. 4 and Rs. 11 respectively.

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