Solve the system of equations using a matrix%0D%0A%0D%0A1.5x−0.5y=−3.5%0D%0A1.5%0D%0A%0D%0A−%0D%0A0.5%0D%0A%0D%0A=%0D%0A−%0D%0A3.5%0D%0A %0D%0A1.5x+1.5y=−7.5%0D%0A1.5%0D%0A%0D%0A+%0D%0A1.5%0D%0A%0D%0A=%0D%0A−%0D%0A7.5%0D%0A (1 point)%0D%0AResponses%0D%0A%0D%0A[32]%0D%0A[%0D%0A3%0D%0A2%0D%0A]%0D%0Athe 2 by 1 column matrix 3 2%0D%0A%0D%0A[−2−3]%0D%0A[%0D%0A−%0D%0A2%0D%0A−%0D%0A3%0D%0A]%0D%0Athe 2 by 1 column matrix row 1 negative 2 row 2 negative 3%0D%0A%0D%0A[−3−2]%0D%0A[%0D%0A−%0D%0A3%0D%0A−%0D%0A2%0D%0A]%0D%0Athe 2 by 1 column matrix row 1 negative 3 row 2 negative 2%0D%0A%0D%0Ano solution
To solve the system of equations using a matrix, we can rewrite the system in matrix form:
[A][X] = [B]
where [A] is the coefficient matrix, [X] is the variable matrix, and [B] is the constant matrix.
The coefficient matrix [A] is:
1.5 -0.5
1.5 1.5
The variable matrix [X] is:
x
y
The constant matrix [B] is:
-3.5
-7.5
So, the system becomes:
1.5x - 0.5y = -3.5
1.5x + 1.5y = -7.5
Now, we can solve this system by finding the inverse of [A] and multiplying it with [B]:
[X] = [A]^-1 * [B]
Let's calculate the inverse of [A]:
1.5 -0.5 1 -1
1.5 1.5 = -1 1
Now, let's multiply the inverse of [A] with [B]:
[X] = [-1 -1] * [-3.5]
[1 1] [-7.5]
[X] = [(-1 * -3.5) + (-1 * -7.5)]
[(1 * -3.5) + (1 * -7.5)]
[X] = [3.5 + 7.5]
[-3.5 - 7.5]
[X] = [11]
[-11]
So, the solution to the system of equations is: x = 11, y = -11.
To solve the system of equations using a matrix, we can represent the system in matrix form as follows:
[1.5 -0.5 | -3.5]
[1.5 1.5 | -7.5]
To find the solutions, we will perform row operations on the augmented matrix until we reach row echelon form or reduced row echelon form.
Let's start with the first row:
1. Multiply the first row by 1/1.5 to simplify the leading coefficient:
[1 -1/3 | -7/2]
[1.5 1.5 | -7.5]
2. Subtract 1.5 times the first row from the second row to eliminate the coefficient in the second column:
[1 -1/3 | -7/2]
[0 2 | 4]
3. Multiply the second row by 1/2 to simplify the leading coefficient:
[1 -1/3 | -7/2]
[0 1 | 2]
4. Add (1/3) times the second row to the first row to eliminate the coefficient in the first column:
[1 0 | -11/2]
[0 1 | 2]
We have reached the row echelon form. Now, we can read the solutions directly from the augmented matrix:
x = -11/2
y = 2
Therefore, the solution to the system of equations is x = -11/2 and y = 2.
To solve the system of equations using a matrix, we can represent the system as an augmented matrix and perform row operations to find the solution.
The given system is:
1.5x - 0.5y = -3.5
1.5x + 1.5y = -7.5
Writing this as an augmented matrix:
[ 1.5 -0.5 | -3.5 ]
[ 1.5 1.5 | -7.5 ]
Now, let's perform row operations to solve the matrix.
1. Subtracting the first row from the second row (R2 - R1):
[ 1.5 -0.5 | -3.5 ]
[ 0 2 | -4 ]
2. Dividing the second row by 2 (R2/2):
[ 1.5 -0.5 | -3.5 ]
[ 0 1 | -2 ]
3. Adding 0.5 times the second row to the first row (R1 + 0.5R2):
[ 1.5 0 | -4.5 ]
[ 0 1 | -2 ]
At this point, the matrix is in row-echelon form.
4. Divide the first row by 1.5 (R1/1.5):
[ 1 0 | -3 ]
[ 0 1 | -2 ]
Therefore, the solution to the system of equations is:
x = -3
y = -2
Hence, the correct response is:
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