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`
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.
`A_11 = (-1)^{1 + 1}[(-6,5),(9,-20)] = (-1)^2[120 - 45]`
= `1 xx 75 = 75`
`A_12 = (-1)^{1 + 2}[(4,5),(6,-20)] = (-1)^3[-80 - 30]`
= `-1 xx (-110) = 110`
`A_13 = (-1)^{1 + 3}[(4,-6),(6,9)] = (-1)^4[36 + 36]`
= `1 xx 72 = 72`
`A_21 = (-1)^{2 + 1}[(3,10),(9,-20)] = (-1)^3[-60 - 90]`
= `-1 xx (-150) = 150`
`A_22 = (-1)^{2 + 2}[(2,10),(6,-20)] = (-1)^4[-40 - 60]`
= `1 xx (-100) = -100`
`A_23 = (-1)^{2 + 3}[(2,3),(6,9)] = (-1)^5[18 - 18] = 0`
`A_31 = (-1)^{3 + 1}[(3,10),(-6,5)] = (-1)^4[15 + 60]`
= `1 xx 75 = 75`
`A_32 = (-1)^{3 + 2}[(2,10),(4,5)] = (-1)^5[10 - 40]`
= `-1 xx (-30) = 30`
`A_33 = (-1)^{3 + 3}[(2,3),(4,-6)] = (-1)^6[-12 - 12]`
= `1 xx (-24) = -24`
∴ adj A = `[(75,110,72),(150,-100,0),(75,30,-24)]`
= `[(75,150,75),(110,-100,30),(72,0,-24)]`
|A| = `a_11A_11 + a_12A_12 + a_13A_13`
= `2 xx 75 + 3 xx 110 + 10 xx 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 `|[2x,5],[8,x]|=|[6,-2],[7,3]|`, write the value of x.
Evaluate the following determinant:
\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}3 & x \\ x & 1\end{vmatrix} = \begin{vmatrix}3 & 2 \\ 4 & 1\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\left( 2^x + 2^{- x} \right)^2 & \left( 2^x - 2^{- x} \right)^2 & 1 \\ \left( 3^x + 3^{- x} \right)^2 & \left( 3^x - 3^{- x} \right)^2 & 1 \\ \left( 4^x + 4^{- x} \right)^2 & \left( 4^x - 4^{- x} \right)^2 & 1\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
Solve the following determinant equation:
If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
Find the value of x if the area of ∆ is 35 square cms with vertices (x, 4), (2, −6) and (5, 4).
x − 2y = 4
−3x + 5y = −7
Prove that :
Prove that :
2x − y = 17
3x + 5y = 6
Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]
Evaluate \[\begin{vmatrix}4785 & 4787 \\ 4789 & 4791\end{vmatrix}\]
Write the value of \[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .\]
Write the cofactor of a12 in the following matrix \[\begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix} .\]
If |A| = 2, where A is 2 × 2 matrix, find |adj A|.
Find the maximum value of \[\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}\]
If \[∆_1 = \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{vmatrix}, ∆_2 = \begin{vmatrix}1 & bc & a \\ 1 & ca & b \\ 1 & ab & c\end{vmatrix},\text{ then }\]}
Solve the following system of equations by matrix method:
x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1
A total amount of ₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and \[8\frac{1}{2}\] % respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0
x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
If \[\begin{bmatrix}1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}\], find x, y and z.
Write the value of `|(a-b, b- c, c-a),(b-c, c-a, a-b),(c-a, a-b, b-c)|`
Solve the following by inversion method 2x + y = 5, 3x + 5y = −3
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
Using determinants, find the equation of the line joining the points (1, 2) and (3, 6).
`abs ((("b" + "c"^2), "a"^2, "bc"),(("c" + "a"^2), "b"^2, "ca"),(("a" + "b"^2), "c"^2, "ab")) =` ____________.
The number of values of k for which the linear equations 4x + ky + 2z = 0, kx + 4y + z = 0 and 2x + 2y + z = 0 possess a non-zero solution is
If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then
If a, b, c are non-zero real numbers and if the system of equations (a – 1)x = y + z, (b – 1)y = z + x, (c – 1)z = x + y, has a non-trivial solution, then ab + bc + ca equals ______.
