Please help!
x-2y+z=7
2x+y-z=0
3x+2y-2z=-2
a. Solve the above system of equations using Gaussian Elimination or Gauss-Jordan Elimination. You must show row operations.
b. Solve the above system of equations using Cramer's Rule.
its a simple matrix problem. ti-89 does it for you.
how do a graph on a Ti-83 for an ellipse center at origin
This is my problem
I need help with.
(7,-8); 3x+5y=6
{7x+4y=11
5x+6y=2
a. To solve the given system of equations using Gaussian Elimination or Gauss-Jordan Elimination, we will use a sequence of row operations to transform the system into an equivalent system that is simpler to solve.
Step 1: Write the system of equations in matrix form:
[1 -2 1 | 7]
[2 1 -1 | 0]
[3 2 -2 |-2]
Step 2: Apply row operations to eliminate the x-terms below the first equation's pivot. In this case, the pivot is the first element in the first row (which is 1).
Row 2 = Row 2 - 2Row 1
Row 3 = Row 3 - 3Row 1
The updated matrix becomes:
[1 -2 1 | 7]
[0 5 -3 |-14]
[0 8 -5 |-23]
Step 3: Now, we need to eliminate the x-terms below the second equation's pivot. In this case, the pivot is the second element in the second row (which is 5).
Row 3 = Row 3 - (8/5)Row 2
The updated matrix becomes:
[1 -2 1 | 7]
[0 5 -3 |-14]
[0 0 1 | -9]
Step 4: At this point, we have an upper triangular matrix. We can now back-substitute to find the values of the variables.
From the third equation, we get z = -9.
Substituting this back into the second equation, we get:
5y - 3(-9) = -14 => 5y + 27 = -14 => 5y = -41 => y = -41/5.
Finally, substituting these values into the first equation:
x - 2(-41/5) + (-9) = 7 => x + 82/5 - 45/5 = 7 => x + 37/5 = 7 => x = 7 - 37/5 = -8/5.
Therefore, the solution to the given system of equations using Gaussian Elimination is x = -8/5, y = -41/5, and z = -9.
b. To solve the given system of equations using Cramer's Rule, we will use determinants.
Step 1: Calculate the determinant of the coefficient matrix (D):
D = |1 -2 1|
|2 1 -1|
|3 2 -2|
D = (1 * 1 * -2) + (2 * -1 * 3) + (1 * 2 * 2) - (-2 * 1 * 3) - (1 * -1 * 1) - (2 * 2 * -2)
= -2 - 6 + 4 + 6 + 1 - 8
= -5
Step 2: Calculate the determinant of the x-variable matrix (Dx):
Dx = |7 -2 1|
|0 1 -1|
|-2 2 -2|
Dx = (7 * 1 * -2) + (0 * -1 * -2) + (-2 * 2 * 1) - (-2 * 1 * -2) - (7 * -1 * -2) - (0 * 2 * 1)
= -14 + 0 - 4 - 4 + 14 + 0
= -8
Step 3: Calculate the determinant of the y-variable matrix (Dy):
Dy = |1 7 1|
|2 0 -1|
|3 -2 -2|
Dy = (1 * 0 * -2) + (2 * -1 * 3) + (1 * -2 * -2) - (-2 * 0 * 3) - (1 * -1 * -2) - (2 * -2 * -2)
= 0 - 6 + 4 + 0 + 1 - 8
= -9
Step 4: Calculate the determinant of the z-variable matrix (Dz):
Dz = |1 -2 7|
|2 1 0|
|3 2 -2|
Dz = (1 * 1 * -2) + (2 * 0 * 3) + (1 * 2 * 7) - (-2 * 1 * 3) - (1 * 0 * 7) - (2 * 2 * -2)
= -2 + 0 + 14 + 6 + 0 -8
= 10
Step 5: Solve for x, y, and z:
x = Dx / D = -8 / -5 = 8/5
y = Dy / D = -9 / -5 = 9/5
z = Dz / D = 10 / -5 = -2
Therefore, the solution to the given system of equations using Cramer's Rule is x = 8/5, y = 9/5, and z = -2.