Advertisements
Advertisements
Question
Solve the following systems of linear equations by Gaussian elimination method:
2x – 2y + 3z = 2, x + 2y – z = 3, 3x – y + 2z = 1
Advertisements
Solution
The matrix form of the above equations is
`[(2, -2, 3),(1, 2, -1),(3, -1, 2)][(x),(y),(z)] = [(2),(3),(1)]`
(i.e) AX = B
The augment matrix (A, B) is
(A, B) = `[(2, -2, 3, 2),(1, 2, -1, 3),(3, -1, 2, 1)] ˜ [(1, 2, -1, 3),(2, -2, 3, 2),(3, -1, 2, 1)] "R"_1 ↔ "R"_2`
` ˜ [(1, 2, -1, 3),(0, -6, 5, -4),(0, -7, 5, -8)] "R"_2 -> "R"_2 - 2"R"_1; "R"_3 -> "R"_3 - 3"R"_1`
` ˜ [(1, 2, -1, 3),(0, -6, 5, -4),(0, 0, -5, -20)] "R"_3 -> 6"R"_3 - 7R"_2`
The above matrix is in echelon form.
Now writing the equivalent equations
`[(1, 2, -1),(0, -6, 5),(0, 0, -5)][(x),(y),(z)] = [(3),(-4),(-20)]`
(i.e) x + 2 – z = 3
– 6y + 5z = – 4
– 5z = – 20
From (3)
⇒ z = `(-20)/(-5)` = 4
Substituting z = 4 in (2) we get
– 6y + 20 = – 4
⇒ – 6y = – 4 – 20 = – 24
⇒ y = 4
Substituting z = 4 and y = 4 in (1) we get
x + 8 – 4 = 3
⇒ x + 4 = 3
⇒ x = 3 – 4 = – 1
So, x = – 1, y = 4, z = 4
APPEARS IN
RELATED QUESTIONS
Solve the following system of linear equations by matrix inversion method:
2x + 5y = – 2, x + 2y = – 3
Solve the following system of linear equations by matrix inversion method:
2x – y = 8, 3x + 2y = – 2
Solve the following system of linear equations by matrix inversion method:
2x + 3y – z = 9, x + y + z = 9, 3x – y – z = – 1
If A = `[(-5, 1, 3),(7, 1, -5),(1, -1, 1)]` and B = `[(1, 1, 2),(3, 2, 1),(2, 1, 3)]`, Find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2
A man is appointed in a job with a monthly salary of certain amount and a fixed amount of annual increment. If his salary was ₹ 19,800 per month at the end of the first month after 3 years of service and ₹ 23,400 per month at the end of the first month after 9 years of service, find his starting salary and his annual increment. (Use matrix inversion method to solve the problem.)
Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method
The prices of three commodities A, B and C are ₹ x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B . Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn ₹ 15,000, ₹ 1,000 and ₹ 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)
Solve the following systems of linear equations by Cramer’s rule:
5x – 2y + 16 = 0, x + 3y – 7 = 0
Solve the following systems of linear equations by Cramer’s rule:
3x + 3y – z = 11, 2x – y + 2z = 9, 4x + 3y + 2z = 25
In a competitive examination, one mark is awarded for every correct answer while `1/4` mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly? (Use Cramer’s rule to solve the problem).
An amount of ₹ 65,000 is invested in three bonds at the rates of 6%, 8% and 9% per annum respectively. The total annual income is ₹ 4,800. The income from the third bond is ₹ 600 more than that from the second bond. Determine the price of each bond. (Use Gaussian elimination method.)
A boy is walking along the path y = ax2 + bx + c through the points (– 6, 8), (– 2, – 12), and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)
Choose the correct alternative:
If A = `[(costheta, sintheta),(-sintheta, costheta)]` and A(adj A) = `[("k", 0),(0, "k")]`, then k =
Choose the correct alternative:
If adj A = `[(2, 3),(4, 1)]` and adj B = `[(1, -2),(-3, 1)]` then adj (AB) is
Choose the correct alternative:
If 0 ≤ θ ≤ π and the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ) x – y + z = 0, (sin θ) x + y + z = 0 has a non-trivial solution then θ is
Choose the correct alternative:
The augmented matrix of a system of linear equations is `[(1, 2, 7, 3),(0, 1, 4, 6),(0, 0, lambda - 7, mu + 7)]`. This system has infinitely many solutions if
