Solve the system of equations using matrices. If system has no solution say its inconsistent.
6x + y = 5
3x - y + z - w = 8
z + w = 2
Solve the system of equations using Cramers Rule if applicable.
4x + 2y = 8/5
5x - 5y = 5
1.
Rewrite system in augmented matrix form:
0 6 1 0 5
1 3 -1 1 8
1 0 0 1 2
Solve by Gaussian elimination.
With 3 equations and 4 unknowns (w,x,y,z), the solution will be in terms of one of the unknowns and not completely numeric. It could also be inconsistent.
2.
Find the determinant of D=
4 2
5 -5
D=4*(-5)-5*2=-30
Dx (obtained by replacing the x column with the column vector)
8/5 2
5 -5
= (8/5)*(-5) - 5*2
=-18
Dy =
4 8/5
5 5
= 4*5 - 5*(8/5)
= 12
x=Dx/D=-18/(-30)=3/5
y=Dy/D=12/(-30)=-2/5
Check: 4x+2y = 8/5
5x-5y = 5
OK
Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set−builder notation to write the solution set. If a system has no solution, state this.
2x − y = 4,
5x − y = 13
To solve the system of equations using matrices, we can represent the system of equations in matrix form. Let's call the coefficient matrix A, the variable matrix X, and the constant matrix B:
A =
| 6 1 0 0 |
| 3 -1 1 -1 |
| 0 0 1 1 |
X =
| x |
| y |
| z |
| w |
B =
| 5 |
| 8 |
| 2 |
To solve for X, we can use the matrix equation AX = B. To find X, we need to find the inverse of matrix A. If the inverse exists, we can find X by multiplying both sides of the equation by the inverse of A:
X = A^(-1) * B
Once we have the inverse matrix, we can multiply it with the constant matrix B to obtain the solution for X.
However, before proceeding, we need to check if the matrix A is invertible. If the determinant of A is zero, then A is not invertible, and the system of equations has no solution.
The determinant of matrix A is:
| 6 1 0 0 |
| 3 -1 1 -1 |
| 0 0 1 1 |
| 0 0 0 0 |
To find the determinant, we can use the Laplace expansion along the first row:
det(A) = 6 * det(
| -1 1 -1 |
| 0 1 1 |
| 0 0 0 |
)
- 1 * det(
| 3 1 -1 |
| 0 1 1 |
| 0 0 0 |
)
= 6 * (-1 * det(
| 1 1 |
| 1 0 |
) )
- 1 * (3 * det(
| 1 1 |
| 1 0 |
) )
= 6 * (-1 * (1 * 0 - 1 * 1))
- 1 * (3 * (1 * 0 - 1 * 1))
= 6 * (-1 * (-1))
- 1 * (3 * (-1))
= 6
Since the determinant is 6, which is not equal to zero, we can continue to find the solution.
Next, we need to find the inverse of matrix A. The formula to find the inverse of a 3x3 matrix is:
A^(-1) = (adj(A))/det(A)
Where adj(A) is the adjugate of matrix A. The adjugate of matrix A is the transpose of the cofactor matrix.
First, let's find the cofactor matrix of A.
Cofactor matrix of A =
| -1 1 -1 |
| 1 0 0 |
| 0 0 0 |
Next, let's find the transpose of the cofactor matrix of A.
Transpose of Cofactor matrix of A =
| -1 1 0 |
| 1 0 0 |
| -1 0 0 |
Now we have the adjugate of matrix A.
adj(A) =
| -1 1 0 |
| 1 0 0 |
| -1 0 0 |
To find the inverse of A, we can divide the adjugate of A by the determinant of A.
A^(-1) = adj(A) / det(A)
= (1/6) *
| -1 1 0 |
| 1 0 0 |
| -1 0 0 |
Now we can multiply the inverse matrix A^(-1) with the constant matrix B to find the solution for X.
X = A^(-1) * B
= (1/6) *
| -1 1 0 |
| 1 0 0 |
| -1 0 0 |
*
| 5 |
| 8 |
| 2 |
The result will give us the values of x, y, z, and w, which are the solutions to the system of equations.
Now let's solve the second system of equations using Cramer's Rule. Cramer's Rule allows us to find the solution for the variables of a system of equations by using determinants.
The given system of equations is:
4x + 2y = 8/5
5x - 5y = 5
To use Cramer's Rule, we need to find the determinants.
The determinant of the coefficient matrix, D, is:
D =
| 4 2 |
| 5 -5 |
D = (4 * -5) - (2 * 5)
= -20 - 10
= -30
The determinants of the matrices formed by replacing the x-coefficients and y-coefficients with the constant terms, Dx and Dy, are:
Dx =
| 8/5 2 |
| 5 -5 |
Dx = (8/5 * -5) - (2 * 5)
= -40/5 - 10
= -8 - 10
= -18
Dy =
| 4 8/5 |
| 5 5 |
Dy = (4 * 5) - (8/5 * 5)
= 20 - 40/5
= 20 - 8
= 12
Now we can find the solution for x and y using Cramer's Rule:
x = Dx / D
= -18 / -30
= 3/5
y = Dy / D
= 12 / -30
= -2/5
Therefore, the solution for the system of equations is x = 3/5 and y = -2/5.