Advertisements
Advertisements
प्रश्न
Solve the system of the following equations:
`2/x+3/y+10/z = 4`
`4/x-6/y + 5/z = 1`
`6/x + 9/y - 20/x = 2`
Advertisements
उत्तर
The given equation,
`2/x + 3/y + 10/z = 4`
`4/x - 6/y + 5/z = 1`
`6/x + 9/y - 20/z = 2`
Let, `1/x` = u, `1/y` = v, `1/z` = w
∴ 2u + 3v + 10w = 4
4u − 6v + 5w = 1
6u + 9v − 20w = 2
This can be written as AX = B, where
A = `[(2,3,10),(4,-6,5),(6,9,-20)]`, X = `[(u),(v),(w)]`, B = `[(4),(1),(2)]`
The element Aij is the cofactor of aij.
A11 = `(-1)^(1 + 1)[(-6,5),(9,-20)]`
= (−1)2 [120 − 45]
= 1 × 75
= 75
A12 = `(-1)^(1 + 2)[(4,5),(6,-20)]`
= (−1)3 [−80 − 30]
= −1 × (−110)
= 110
A13 = `(-1)^(1 + 3)[(4,-6),(6,9)]`
= (−1)4 [36 + 36]
= 1 × 72
= 72
A21 = `(-1)^(2 + 1)[(3,10),(9,-20)]`
= (−1)3 [−60 − 90]
= −1 × (−150)
= 150
A22 = `(-1)^(2 + 2)[(2,10),(6,-20)]`
= (−1)4 [−40 − 60]
= 1 × (−100)
= −100
A23 = `(-1)^(2 + 3)[(2,3),(6,9)]`
= (−1)5 [18 - 18]
= 0
A31 = `(-1)^(3 + 1)[(3,10),(-6,5)]`
= (−1)4 [15 + 60]
= 1 × 75
= 75
A32 = `(-1)^(3 + 2)[(2,10),(4,5)]`
= (−1)5 [10 − 40]
= −1 × (−30)
= 30
A33 = `(-1)^(3 + 3)[(2,3),(4,-6)]`
= (−1)6 [−12 − 12]
= 1 × (−24)
= −24
∴ adj A = `[(75,110,72),(150,-100,0),(75,30,-24)]`
= `[(75,150,75),(110,-100,30),(72,0,-24)]`
|A| = a11A11 + a12A12 + a13A13
= 2 × 75 + 3 × 110 + 10 × 72
= 150 + 330 + 720
= 1200
A−1 = `1/|A|` (adj A)
= `1/1200[(75,150,75),(110,-100,30),(72,0,-24)]`
X = A−1B
= `1/1200[(75,150,75),(110,-100,30),(72,0,-24)][(4),(1),(2)]`
`[(u),(v),(w)] = 1/1200[(300 + 150 + 150),(440 - 100 + 60),(288 + 0 - 48)]`
= `1/12000 [(600),(400),(240)]`
= `[(1/2),(1/3),(1/5)]`
∴ u = `1/2`, v = `1/3`, w = `1/5`
⇒ x = `1/u` = 2, y = `1/v` = 3, z = `1/w` = 5
Hence, the solutions of the system of equations are x = 2, y = 3, z = 5.
APPEARS IN
संबंधित प्रश्न
If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A−1. Using A−1 solve the system of equations:
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = –3
Show that
\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]
For what value of x the matrix A is singular?
\[A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .\]
Prove the following identities:
\[\begin{vmatrix}y + z & z & y \\ z & z + x & x \\ y & x & x + y\end{vmatrix} = 4xyz\]
Prove the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
Solve the following determinant equation:
Find the value of \[\lambda\] so that the points (1, −5), (−4, 5) and \[\lambda\] are collinear.
Prove that :
Prove that :
Prove that
3x + y = 19
3x − y = 23
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0
Write the value of the determinant
If w is an imaginary cube root of unity, find the value of \[\begin{vmatrix}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{vmatrix}\]
Find the value of the determinant \[\begin{vmatrix}243 & 156 & 300 \\ 81 & 52 & 100 \\ - 3 & 0 & 4\end{vmatrix} .\]
Write the value of \[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .\]
If \[A = \begin{bmatrix}\cos\theta & \sin\theta \\ - \sin\theta & \cos\theta\end{bmatrix}\] , then for any natural number, find the value of Det(An).
The value of the determinant
Let \[A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}\]
If \[\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} = 16\] , then the value of \[\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix}\] is
The value of \[\begin{vmatrix}1 & 1 & 1 \\ {}^n C_1 & {}^{n + 2} C_1 & {}^{n + 4} C_1 \\ {}^n C_2 & {}^{n + 2} C_2 & {}^{n + 4} C_2\end{vmatrix}\] is
Solve the following system of equations by matrix method:
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Show that the following systems of linear equations is consistent and also find their solutions:
6x + 4y = 2
9x + 6y = 3
Show that the following systems of linear equations is consistent and also find their solutions:
x − y + z = 3
2x + y − z = 2
−x −2y + 2z = 1
Show that the following systems of linear equations is consistent and also find their solutions:
2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3
The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.
If \[\begin{bmatrix}1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ - 1 \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}\] , find x, y and z.
Solve the following for x and y: \[\begin{bmatrix}3 & - 4 \\ 9 & 2\end{bmatrix}\binom{x}{y} = \binom{10}{ 2}\]
Let \[X = \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}, A = \begin{bmatrix}1 & - 1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}3 \\ 1 \\ 4\end{bmatrix}\] . If AX = B, then X is equal to
Show that if the determinant ∆ = `|(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)|` = 0, then sinθ = 0 or `1/2`.
If `alpha, beta, gamma` are in A.P., then `abs (("x" - 3, "x" - 4, "x" - alpha),("x" - 2, "x" - 3, "x" - beta),("x" - 1, "x" - 2, "x" - gamma)) =` ____________.
The value of λ, such that the following system of equations has no solution, is
`2x - y - 2z = - 5`
`x - 2y + z = 2`
`x + y + lambdaz = 3`
If `|(x + 1, x + 2, x + a),(x + 2, x + 3, x + b),(x + 3, x + 4, x + c)|` = 0, then a, b, care in
