Advertisements
Advertisements
Question
2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11
Advertisements
Solution
Given: 2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11
\[D = \begin{vmatrix}2 & - 3 & - 4 \\ - 2 & 5 & - 1 \\ 3 & - 1 & 5\end{vmatrix}\]
\[ = 2(25 - 1) + 3( - 10 + 3) - 4(2 - 15)\]
\[ = 2(24) + 3( - 7) - 4( - 13)\]
\[ = 79\]
\[ D_1 = \begin{vmatrix}29 & - 3 & - 4 \\ - 15 & 5 & - 1 \\ - 11 & - 1 & 5\end{vmatrix}\]
\[ = 29(25 - 1) + 3( - 75 - 11) - 4(15 + 55)\]
\[ = 29(24) + 3( - 86) - 4(70)\]
\[ = 158\]
\[ D_2 = \begin{vmatrix}2 & 29 & - 4 \\ - 2 & - 15 & - 1 \\ 3 & - 11 & 5\end{vmatrix}\]
\[ = 2( - 75 - 11) - 29( - 10 + 3) - 4(22 + 45)\]
\[ = 2( - 86) - 29( - 7) - 4(67)\]
\[ = - 237\]
\[ D_3 = \begin{vmatrix}2 & - 3 & 29 \\ - 2 & 5 & - 15 \\ 3 & - 1 & - 11\end{vmatrix}\]
\[ = 2( - 55 - 15) + 3(22 + 45) + 29(2 - 15)\]
\[ = 2( - 70) + 3(67) + 29( - 13)\]
\[ = - 316\]
Now,
\[x = \frac{D_1}{D} = \frac{158}{79} = 2\]
\[y = \frac{D_2}{D} = \frac{- 237}{79} = - 3\]
\[z = \frac{D_3}{D} = \frac{- 316}{79} = - 4\]
\[ \therefore x = 2, y = - 3\text{ and }z = - 4\]
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
x + 3y = 5
2x + 6y = 8
Solve the system of linear equations using the matrix method.
2x – y = –2
3x + 4y = 3
Solve the system of linear equations using the matrix method.
5x + 2y = 3
3x + 2y = 5
Solve the system of linear equations using the matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}\]
\[\begin{vmatrix}b + c & a & a \\ b & c + a & b \\ c & c & a + b\end{vmatrix} = 4abc\]
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:
(2, 7), (1, 1) and (10, 8)
Find the area of the triangle with vertice at the point:
(0, 0), (6, 0) and (4, 3)
Prove that :
Prove that :
5x + 7y = − 2
4x + 6y = − 3
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
If w is an imaginary cube root of unity, find the value of \[\begin{vmatrix}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{vmatrix}\]
If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.
For what value of x is the matrix \[\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}\] singular?
Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 0
Solve the following system of equations by matrix method:
x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1
Solve the following system of equations by matrix method:
2x + 6y = 2
3x − z = −8
2x − y + z = −3
Show that the following systems of linear equations is consistent and also find their solutions:
5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5
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.
Two factories decided to award their employees for three values of (a) adaptable tonew techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ x, ₹ y and ₹ z per person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then
i) represent the above situation by matrix equation and form linear equation using matrix multiplication.
ii) Solve these equation by matrix method.
iii) Which values are reflected in the questions?
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.
If \[A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}\] ,find A–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7.
If A = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A−1X = B.
Show that if the determinant ∆ = `|(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)|` = 0, then sinθ = 0 or `1/2`.
In system of equations, if inverse of matrix of coefficients A is multiplied by right side constant B vector then resultant will be?
If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then
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
For what value of p, is the system of equations:
p3x + (p + 1)3y = (p + 2)3
px + (p + 1)y = p + 2
x + y = 1
consistent?
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
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 ______.
