Advertisements
Advertisements
Question
Solve the following system of equations by the method of reduction:
x + y + z = 6, y + 3z = 11, x + z = 2y.
Solution
Given:
x + y + z = 6
y + 3z = 11
x + z = 2y
`[(1, 1, 1), (0, 1, 3), (1, -2, 1)] [(x), (y), (z)] = [(6), (11), (0)]`
using R3 → R3 – R1
we get `[(1, 1, 1), (0, 1, 3),(0, -3, 0)] [(x), (y), (z)] = [(6), (11), (-6)]`
using R3 → R3 + 3R2
we get `[(1, 1, 1), (0, 1, 3),(0, 0, 9)] [(x), (y), (z)] = [(6), (11), (27)]`
∴ `[(x + y + z),(0 + y + 3z),(0 + 0 + 9z)] = [(6), (11), (27)]`
Equating, we get
x + y + z = 6 ...(1)
y + 3z = 11 ...(2)
9z = 27 ...(3)
z = `27/9` ...[from eq. (3)]
z = 3
put this value of z in equation (2), we get
y + 3(3) = 11
y + 9 = 11
y = 11 − 9
y = 2
put the values of y and z in equation (1), we get
x + y + z = 6
x + 2 + 3 = 6
x + 5 = 6
x = 6 − 5
x = 1
APPEARS IN
RELATED QUESTIONS
Solve the following equations by the reduction method.
2x + y = 5, 3x + 5y = – 3
Solve the following equations by the reduction method.
3x – y = 1, 4x + y = 6
Solve the following equations by inversion method:
x + y = 4, 2x - y = 5
Solve the following equations by the method of inversion:
x + y+ z = 1, 2x + 3y + 2z = 2,
ax + ay + 2az = 4, a ≠ 0.
Solve the following equations by the method of inversion:
x + y + z = - 1, y + z = 2, x + y - z = 3
Express the following equations in matrix form and solve them by the method of reduction:
`x + y = 1, y + z = 5/3, z + x 4/33`.
Express the following equations in matrix form and solve them by the method of reduction:
2x - y + z = 1, x + 2y + 3z = 8, 3x + y - 4z = 1.
Express the following equations in matrix form and solve them by the method of reduction:
x + 2y + z = 8, 2x + 3y - z = 11, 3x - y - 2z = 5.
The cost of 4 pencils, 3 pens, and 2 books is ₹ 150. The cost of 1 pencil, 2 pens, and 3 books is ₹ 125. The cost of 6 pencils, 2 pens, and 3 books is ₹ 175. Find the cost of each item by using matrices.
The sum of three numbers is 6. Thrice the third number when added to the first number, gives 7. On adding three times the first number to the sum of second and third numbers, we get 12. Find the three number by using matrices.
An amount of ₹ 5000 is invested in three types of investments, at interest rates 6%, 7%, 8% per annum respectively. The total annual income from these investments is ₹ 350. If the total annual income from the first two investments is ₹ 70 more than the income from the third, find the amount of each investment using matrix method.
Solve the following equations by the method of inversion:
2x + 3y = - 5, 3x + y = 3
Express the following equations in matrix form and solve them by the method of reduction:
x + 3y + 2z = 6,
3x − 2y + 5z = 5,
2x − 3y + 6z = 7
Solve the following equation by the method of inversion.
2x – y + z = 1,
x + 2y + 3z = 8,
3x + y – 4z = 1
Solve the following equations by method of inversion.
x + y + z = 1, x – y + z = 2 and x + y – z = 3
Express the following equations in matrix form and solve them by method of reduction.
x + y + z = 1, 2x + 3y + 2z = 2 and x + y + 2z = 4
The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.
The sum of the cost of one Economic book, one Co-operation book and one account book is ₹ 420. The total cost of an Economic book, 2 Co-operation books and an Account book is ₹ 480. Also the total cost of an Economic book, 3 Co-operation books and 2 Account books is ₹ 600. Find the cost of each book using matrix method.
Find x, y, z, if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, - 2),(1, 3)]} [(2),(1)] = [(x - 1),(y + 1),(2z)]`
Solve the following :
Two farmers Shantaram and Kantaram cultivate three crops rice, wheat and groundnut. The sale (in Rupees) of these crops by both the farmers for the month of April and May 2016 is given below,
| April 2016 (in ₹.) | |||
| Rice | Wheat | Groundnut | |
| Shantaram | 15000 | 13000 | 12000 |
| Kantaram | 18000 | 15000 | 8000 |
| May 2016 (in ₹.) | |||
| Rice | Wheat | Groundnut | |
| Shantaram | 18000 | 15000 | 12000 |
| Kantaram | 21000 | 16500 | 16000 |
Find : the increase in sale from April to May for every crop of each farmer.
Solve the following equations by method of inversion : x + y – z = 2, x – 2y + z = 3 and 2x – y – 3z = – 1
Solve the following equations by method of inversion : x – y + z = 4, 2x + y – 3z = 0 , x + y + z = 2
Solve the following equations by method of reduction :
x + 2y - z = 3 , 3x – y + 2z = 1 and 2x – 3y + 3z = 2
Solve the following equations by method of reduction :
x – 3y + z = 2 , 3x + y + z = 1 and 5x + y + 3z = 3
If A2 + 5A + 3I = 0, |A| ≠ 0, then A–1 = ______
State whether the following statement is True or False:
If O(A) = m × n and O(B) = n × p with m ≠ p, then BA exists but AB does not exist.
Complete the following activity.
The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.
Solution: Let the cost of potato, wheat and rice per kg be x, y and z respectively.
Therefore by given conditions,
4x + ( )y + 2( ) = ( )
x + 2y + ( )( ) = ( )
( )x + 2y + 3z = ( )
Matrix form of above equations is,
`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`
R1 ↔ R2
`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`
R2 – 4R1, R3 – 6R1
`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`
`(-1)/5 "R"_2, (-1)/5 "R"_3`
`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`
R3 – 2R2
`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`
By pre multiplying we get,
x + 2y + ( )z = ( ) .....(i)
y + 2z = 24 ......(ii)
–z = ( ) ......(iii)
From (iii), we get, z = ( )
From (ii), we get, y = ( )
From (i), we get, x = ( )
Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.
If A = `[(1, -1, 3), (2, 5, 4)]`, then R1 ↔ R2 and C3 → C3 + 2C2 gives ______
If `[(1, -1, x), (1, x, 1), (x, -1, 1)]` has no inverse, then the real value of x is ______
Adjoint of ______
If A =`[(1, -1), (2, 3)]` and adj (A) = `[(a, b), (-2, 1)]`, then ______
Solve the following system of equations by the method of inversion.
x – y + z = 4, 2x + y – 3z = 0, x + y + z = 2
