Advertisements
Advertisements
प्रश्न
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
Advertisements
उत्तर
Given: 3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20
\[D = \begin{vmatrix}3 & - 1 & 2 \\ 2 & - 1 & 1 \\ 3 & 6 & 5\end{vmatrix}\]
\[3\left( - 5 - 6 \right) + 1\left( 10 - 3 \right) + 2\left( 12 + 3 \right) = 4\]
Since D is non-zero, the system of linear equations is consistent and has a unique solution.
\[ D_1 = \begin{vmatrix}6 & - 1 & 2 \\ 2 & - 1 & 1 \\ 20 & 6 & 5\end{vmatrix}\]
\[ = 6\left( - 5 - 6 \right) + 1\left( 10 - 20 \right) + 2\left( 12 + 20 \right)\]
\[ = - 66 - 10 + 64\]
\[ = - 12\]
\[ D_2 = \begin{vmatrix}3 & 6 & 2 \\ 2 & 2 & 1 \\ 3 & 20 & 5\end{vmatrix}\]
\[ = 3\left( 10 - 20 \right) - 6\left( 10 - 3 \right) + 2\left( 40 - 6 \right)\]
\[ = - 30 - 42 + 68\]
\[ = - 4\]
\[ D_3 = \begin{vmatrix}3 & - 1 & 6 \\ 2 & - 1 & 2 \\ 3 & 6 & 20\end{vmatrix}\]
\[ = 3\left( - 20 - 12 \right) + 1\left( 40 - 6 \right) + 6\left( 12 + 3 \right)\]
\[ = - 96 + 34 + 90\]
\[ = 28\]
Now,
\[x = \frac{D_1}{D} = \frac{- 12}{4} = - 3\]
\[y = \frac{D_2}{D} = \frac{- 4}{4} = - 1\]
\[z = \frac{D_3}{D} = \frac{28}{4} = 7\]
\[ \therefore x = - 3, y = - 1\text{ and }z = 7\]
APPEARS IN
संबंधित प्रश्न
Solve the system of linear equations using the matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
\[\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + a & a \\ 1 & 1 & 1 + a\end{vmatrix} = a^3 + 3 a^2\]
Show that
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
Prove that
Prove that
9x + 5y = 10
3y − 2x = 8
5x − 7y + z = 11
6x − 8y − z = 15
3x + 2y − 6z = 7
Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0
For what value of x, the following matrix is singular?
If the matrix \[\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}\] is singular, find the value of x.
Find the value of the determinant \[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}\].
Write the cofactor of a12 in the following matrix \[\begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix} .\]
If \[\begin{vmatrix}2x & x + 3 \\ 2\left( x + 1 \right) & x + 1\end{vmatrix} = \begin{vmatrix}1 & 5 \\ 3 & 3\end{vmatrix}\], then 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).
If a > 0 and discriminant of ax2 + 2bx + c is negative, then
\[∆ = \begin{vmatrix}a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0\end{vmatrix} is\]
If a, b, c are in A.P., then the determinant
\[\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}\]
If x, y, z are different from zero and \[\begin{vmatrix}1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z\end{vmatrix} = 0\] , then the value of x−1 + y−1 + z−1 is
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
Solve the following system of equations by matrix method:
x + y + z = 3
2x − y + z = − 1
2x + y − 3z = − 9
Show that the following systems of linear equations is consistent and also find their solutions:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
Show that each one of the following systems of linear equation is inconsistent:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.
3x − y + 2z = 0
4x + 3y + 3z = 0
5x + 7y + 4z = 0
Let a, b, c be positive real numbers. The following system of equations in x, y and z
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
If the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx - 12y - 14 = 0 has non-trivial solution, then the value of k is ____________.
A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is
If c < 1 and the system of equations x + y – 1 = 0, 2x – y – c = 0 and – bx+ 3by – c = 0 is consistent, then the possible real values of b are
If a, b, c are non-zeros, then the system of equations (α + a)x + αy + αz = 0, αx + (α + b)y + αz = 0, αx+ αy + (α + c)z = 0 has a non-trivial solution if
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 ______.
If the system of linear equations x + 2ay + az = 0; x + 3by + bz = 0; x + 4cy + cz = 0 has a non-zero solution, then a, b, c ______.
