To solve The system of linear equations by using the Cramer’s rule.
The solution of the given system of equations are ( 5 , 8 , − 2 ) .
Given information:
The given system of equations are { 4 x − 2 y + 3 z = − 2 2 x + 2 y + 5 z = 16 8 x − 5 y − 2 z = 4 .
Formula used:
If { a 1 x + b 1 y + + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3 represents the system of equations then the value of x and y are calculated as x = | d 1 b 1 c 1 d 2 b 2 c 2 d 3 b 3 c 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | = D x D , y = | a 1 d 1 c 1 a 2 d 2 c 2 a 3 d 3 c 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | = D y D and z = | a 1 b 1 d 1 a 2 b 2 d 2 a 3 b 3 d 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | = D z D
Calculation:
Compute the determinant of the matrix as follows,
D = [ 4 − 2 3 2 2 5 8 − 5 − 2 ] = 4 ( − 1 ) | 2 5 − 5 − 2 | + ( − 2 ) ( − 1 ) 3 | 2 5 8 − 2 | + 3 ( − 1 ) 4 | 2 2 8 − 5 | = 4 ( − 4 + 25 ) − ( − 2 ) ( − 4 − 40 ) + 3 ( − 10 − 16 ) = − 82
As the determinant of the equation is nonzero, so the solution of the equations exist.
Compute the value of x and y as follows,
x = | − 2 − 2 3 16 2 5 4 − 5 − 2 | − 82 = − 42 − 104 − 264 − 82 = 5
Compute the value of y as follows,
x = | 4 − 2 3 2 16 5 8 4 − 2 | − 82 = − 208 − 88 − 360 − 82 = 8
Compute the value of z as follows,
z = | 4 − 2 − 2 2 2 16 8 − 5 4 | − 82 = 352 − 240 + 52 − 82 = − 2
Substitute the obtained values in the equation 4 x − 2 y + 3 z = − 2 .
4 ( 5 ) − 2 ( 8 ) + 3 ( − 2 ) = − 2 20 − 16 − 10 = − 2 16 = 16
Substitute the obtained value in the second equation 2 x + 2 y + 5 z = 16 as follows,
2 ( 5 ) + 2 ( 8 ) + 5 ( − 2 ) = 16 10 + 16 − 10 = 16 16 = 16
Substitute the obtained values in the equation 8 x − 5 y − 2 z = 4 .
8 ( 5 ) − 5 ( 8 ) − 2 ( − 2 ) = 4 40 − 40 + 4 = 4 4 = 4
Hence, the obtained solution are correct.
Therefore, the solution of the given system of equations are ( 5 , 8 , − 2 ) .
The solution of the given system of equations are
Given information:
The given system of equations are
Formula used:
If
Calculation:
Compute the determinant of the matrix as follows,
As the determinant of the equation is nonzero, so the solution of the equations exist.
Compute the value of x and y as follows,
Compute the value of y as follows,
Compute the value of z as follows,
Substitute the obtained values in the equation
Substitute the obtained value in the second equation
Substitute the obtained values in the equation
Hence, the obtained solution are correct.
Therefore, the solution of the given system of equations are
Answer to Problem 30E
The solution of the given system of equations are
Explanation of Solution
Given information:
The given system of equations are
Formula used:
If
Calculation:
Compute the determinant of the matrix as follows,
As the determinant of the equation is nonzero, so the solution of the equations exist.
Compute the value of x and y as follows,
Compute the value of y as follows,
Compute the value of z as follows,
Substitute the obtained values in the equation
Substitute the obtained value in the second equation
Substitute the obtained values in the equation
Hence, the obtained solution are correct.
Therefore, the solution of the given system of equations are
Chapter 7 Solutions
PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
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