Solve the system of equations using only the answers below 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
The system of equations is:
1.5x - 0.5y = -3.5
1.5x + 1.5y = -7.5
We can solve this system by using matrix operations.
Let's set up the augmented matrix:
[1.5 -0.5 | -3.5]
[1.5 1.5 | -7.5]
To solve the system using only the provided answers, we can perform row operations on the augmented matrix until we reach one of the provided answer choices.
Using row operations, we can multiply the first row by 2/3 and subtract it from the second row:
[1.5 -0.5 | -3.5]
[0 2 | 1 ]
Now, let's multiply the second row by 1/2:
[1.5 -0.5 | -3.5]
[0 1 | 0.5]
We have now reached the answer choice: the 2 by 1 column matrix [0.5, 0.5].
Therefore, the solution to the system of equations is x = 0.5 and y = 0.5.
To solve the system of equations using the given answer choices, we can substitute the values into the equations and check if they satisfy both equations.
Let's start with the first answer choice: [3 2].
Substituting the values into the equations:
For the first equation: 1.5x - 0.5y = -3.5,
we have: 1.5(3) - 0.5(2) = 4.5 - 1 = 3.5. This does not equal -3.5.
For the second equation: 1.5x + 1.5y = -7.5,
we have: 1.5(3) + 1.5(2) = 4.5 + 3 = 7.5. This does not equal -7.5.
Therefore, this answer choice does not solve the system of equations.
Let's move on to the second answer choice: [-2 -3].
For the first equation: 1.5x - 0.5y = -3.5,
we have: 1.5(-2) - 0.5(-3) = -3 - (-1.5) = -3 + 1.5 = -1.5. This does not equal -3.5.
For the second equation: 1.5x + 1.5y = -7.5,
we have: 1.5(-2) + 1.5(-3) = -3 + (-4.5) = -7.5. This equals -7.5.
Therefore, this answer choice does not solve the system of equations either.
Next, let's try the third answer choice: [-3 -2].
For the first equation: 1.5x - 0.5y = -3.5,
we have: 1.5(-3) - 0.5(-2) = -4.5 + 1 = -3.5. This equals -3.5.
For the second equation: 1.5x + 1.5y = -7.5,
we have: 1.5(-3) + 1.5(-2) = -4.5 - 3 = -7.5. This equals -7.5.
Therefore, this answer choice solves the system of equations.
Hence, the solution to the system is x = -3 and y = -2.
To solve the system of equations, you can use the given matrix as the coefficient matrix on the left-hand side and the right-hand side vector on the right-hand side.
1. Write the system of equations in matrix form:
```
[1.5 -0.5] [x] = [-3.5]
[1.5 1.5] [y] [-7.5]
```
2. Take the inverse of the coefficient matrix on the left-hand side (if it exists). In this case, the coefficient matrix is not invertible because its determinant is 0.
3. Check if the system has a solution by comparing the rank of the coefficient matrix with the augmented matrix [coefficient matrix | right-hand side vector]. If the ranks are different, the system has no solution. Otherwise, it has a solution.
4. Since the coefficient matrix is not invertible and the ranks are different, the system of equations has no solution.
Therefore, the correct answer is "no solution."