Use Cramer's rule to solve the system. 2x + 4y - z = 32 x - 2y + 2z = -5 5x + y + z = 20
A. {( 1, -9, -6)}
B. {( 2, 7, 6)}
C. {( 9, 6, 9)}
D. {( 1, 9, 6)}
d ?
+2 +4 -1
+1 -2 +2 ---> -4+40-1 -10-4-4 = 17 yuuk
+5 +1 +1
now solve for Y not x because your solutions set has all different y values :)
+2 32 -1
+1 -5 +2 ---> --10+320-20 -25-80-32 = 153
+5 20 +1
then
153/17 = ( whew, D
I mean 9 :)
SO I'm right ? :)
Yes, it's D.
Though it's hard to show 3x3 Cramer's rule here. ;u;
Thanks !
To solve the given system of equations using Cramer's rule, we need to find the values of x, y, and z that satisfy all three equations.
Cramer's Rule states that the solution to a system of equations can be found using determinants. To use Cramer's Rule, we need to find the determinants of a few matrices.
Step 1: Set up the matrices
The first matrix, D, is the coefficient matrix of the variables (2x + 4y - z = 32, x - 2y + 2z = -5, 5x + y + z = 20).
D = | 2 4 -1 |
| 1 -2 2 |
| 5 1 1 |
The second matrix, Dx, is obtained by replacing the first column of D with the constants on the right side of the equations.
Dx = |32 4 -1 |
|-5 -2 2 |
|20 1 1 |
The third matrix, Dy, is obtained by replacing the second column of D with the constants.
Dy = | 2 32 -1 |
| 1 -5 2 |
| 5 20 1 |
The fourth matrix, Dz, is obtained by replacing the third column of D with the constants.
Dz = | 2 4 32 |
| 1 -2 -5 |
| 5 1 20 |
Step 2: Calculate the determinants
We need to find the determinants of D, Dx, Dy, and Dz.
Det(D) = |D| = 2(2(-2) - 1(1)) - 4(1(1) - 5(1)) - (-1(5(1) - 5(2)))
= 2(-4) - 4(-4) - (-1(-5))
= -8 + 16 - 5
= 3
Det(Dx) = |Dx| = 32(2(-2) - 1(1)) - 4((-5)(1) - 20(2)) - (-1(-5)(20 - 1))
= 32(-4) - 4(-10 - 40) - (-1(-5)(19))
= -128 - 4(-50) - 5(19)
= -128 + 200 - 95
= -23
Det(Dy) = |Dy| = 2(32(-2) - 1(5)) - 32(1(1) - 5(2)) - (-1(5(20) - 1(5)))
= 2(-64) - 32(-8) - (-1(99))
= -128 + 256 - (-99)
= -128 + 256 + 99
= 227
Det(Dz) = |Dz| = 2(4(20) - 1(1)) - 4(1(5) - 5(2)) - 32(5(1) - 5(1))
= 2(80 - 1) - 4(5 - 10) - 32(5 - 5)
= 2(79) - 4(-5) - 32(0)
= 158 + 20 + 0
= 178
Step 3: Solve for x, y, and z
Using Cramer's Rule, we can find the values of x, y, and z.
x = Dx/D = -23/3 = -7.67 (approx)
y = Dy/D = 227/3 = 75.67 (approx)
z = Dz/D = 178/3 = 59.33 (approx)
Therefore, the solution to the system of equations is approximately (x, y, z) = (-7.67, 75.67, 59.33).
None of the given options match this solution, so the correct answer is not provided.