Advertisements
Advertisements
Question
x+ y = 5
y + z = 3
x + z = 4
Advertisements
Solution
These equations can be written as
x + y + 0z = 5
0x + y + z = 3
x + 0y + z = 4
\[D = \begin{vmatrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1\end{vmatrix}\]
\[ = 1(1 - 0) - 1(0 - 1) + 0(0 - 1)\]
\[ = 1(1) - 1( - 1) + 0\]
\[ = 2\]
\[ D_1 = \begin{vmatrix}5 & 1 & 0 \\ 3 & 1 & 1 \\ 4 & 0 & 1\end{vmatrix}\]
\[ = 5(1 - 0) - 1(3 - 4) + 0(0 - 4)\]
\[ = 5(1) - 1( - 1)\]
\[ = 6\]
\[ D_2 = \begin{vmatrix}1 & 5 & 0 \\ 0 & 3 & 1 \\ 1 & 4 & 1\end{vmatrix}\]
\[ = 1(3 - 4) - 5(0 - 1) + 0(0 - 4)\]
\[ = 1( - 1) - 5( - 1)\]
\[ = 4\]
\[ D_3 = \begin{vmatrix}1 & 1 & 5 \\ 0 & 1 & 3 \\ 1 & 0 & 4\end{vmatrix}\]
\[ = 1(4 - 0) - 1(0 - 3) + 5(0 - 1)\]
\[ = 1(4) - 1( - 3) + 5( - 1)\]
\[ = 2\]
Now,
\[x = \frac{D_1}{D} = \frac{6}{2} = 3\]
\[y = \frac{D_2}{D} = \frac{4}{2} = 2\]
\[z = \frac{D_3}{D} = \frac{2}{2} = 1\]
\[ \therefore x = 3, y = 2\text{ and }z = 1\]
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
Find the value of x, if
\[\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}\]
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:
\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix}\]
\[\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2\]
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
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:
Using determinants, find the value of k so that the points (k, 2 − 2 k), (−k + 1, 2k) and (−4 − k, 6 − 2k) may be collinear.
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Using determinants, find the equation of the line joining the points
(3, 1) and (9, 3)
Prove that :
Prove that
2x − y = 17
3x + 5y = 6
5x − 7y + z = 11
6x − 8y − z = 15
3x + 2y − 6z = 7
Write the value of \[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .\]
If \[A = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}\]. Write the cofactor of the element a32.
If \[∆_1 = \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{vmatrix}, ∆_2 = \begin{vmatrix}1 & bc & a \\ 1 & ca & b \\ 1 & ab & c\end{vmatrix},\text{ then }\]}
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 number of distinct real roots of \[\begin{vmatrix}cosec x & \sec x & \sec x \\ \sec x & cosec x & \sec x \\ \sec x & \sec x & cosec x\end{vmatrix} = 0\] lies in the interval
\[- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}\]
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , then x =
There are two values of a which makes the determinant \[∆ = \begin{vmatrix}1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2a\end{vmatrix}\] equal to 86. The sum of these two values is
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:
3x + y = 19
3x − y = 23
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
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
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, find A−1. Using A−1, solve the system of linear equations x − 2y = 10, 2x − y − z = 8, −2y + z = 7.
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
The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
For the system of equations:
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4
`abs (("a"^2, 2"ab", "b"^2),("b"^2, "a"^2, 2"ab"),(2"ab", "b"^2, "a"^2))` is equal to ____________.
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`
Let the system of linear equations x + y + az = 2; 3x + y + z = 4; x + 2z = 1 have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, –y*) are collinear points, then the sum of absolute values of all possible values of α is ______.
Using the matrix method, solve the following system of linear equations:
`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.
