Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
Solve the system of linear equations using the matrix method.
2x + y + z = 1
x – 2y – z = `3/2`
3y – 5z = 9
Solve the system of linear equations using the matrix method.
x − y + z = 4
2x + y − 3z = 0
x + y + z = 2
Evaluate the following determinant:
\[\begin{vmatrix}x & - 7 \\ x & 5x + 1\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}\]
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:
\[\begin{vmatrix}\sin^2 23^\circ & \sin^2 67^\circ & \cos180^\circ \\ - \sin^2 67^\circ & - \sin^2 23^\circ & \cos^2 180^\circ \\ \cos180^\circ & \sin^2 23^\circ & \sin^2 67^\circ\end{vmatrix}\]
Using properties of determinants prove that
\[\begin{vmatrix}x + 4 & 2x & 2x \\ 2x & x + 4 & 2x \\ 2x & 2x & x + 4\end{vmatrix} = \left( 5x + 4 \right) \left( 4 - x \right)^2\]
Without expanding, prove that
\[\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}\]
Solve the following determinant equation:
If \[\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =\] 0, then using properties of determinants, find the value of \[\frac{a}{x} + \frac{b}{y} + \frac{c}{z}\] , where \[x, y, z \neq\] 0
Find the area of the triangle with vertice at the point:
(−1, −8), (−2, −3) and (3, 2)
Prove that :
3x + ay = 4
2x + ay = 2, a ≠ 0
9x + 5y = 10
3y − 2x = 8
Given: x + 2y = 1
3x + y = 4
x + 2y = 5
3x + 6y = 15
Find the value of the determinant
\[\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}\]
Evaluate \[\begin{vmatrix}4785 & 4787 \\ 4789 & 4791\end{vmatrix}\]
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.
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).
Let \[\begin{vmatrix}x^2 + 3x & x - 1 & x + 3 \\ x + 1 & - 2x & x - 4 \\ x - 3 & x + 4 & 3x\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e\]
be an identity in x, where a, b, c, d, e are independent of x. Then the value of e is
If a, b, c are in A.P., then the determinant
\[\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}\]
The value of the determinant \[\begin{vmatrix}x & x + y & x + 2y \\ x + 2y & x & x + y \\ x + y & x + 2y & x\end{vmatrix}\] is
The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.
A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 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.
Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
`abs ((("b" + "c"^2), "a"^2, "bc"),(("c" + "a"^2), "b"^2, "ca"),(("a" + "b"^2), "c"^2, "ab")) =` ____________.
The number of real value of 'x satisfying `|(x, 3x + 2, 2x - 1),(2x - 1, 4x, 3x + 1),(7x - 2, 17x + 6, 12x - 1)|` = 0 is
Let P = `[(-30, 20, 56),(90, 140, 112),(120, 60, 14)]` and A = `[(2, 7, ω^2),(-1, -ω, 1),(0, -ω, -ω + 1)]` where ω = `(-1 + isqrt(3))/2`, and I3 be the identity matrix of order 3. If the determinant of the matrix (P–1AP – I3)2 is αω2, then the value of α is equal to ______.
