Use a matrix to find the solution to the system of equations.
-8x - 8y= -16
6x - 9y = -108
A. (-6, 8)
B. (6,8)
C. (8, -6)
D. (6, -8)
To solve the system of equations using a matrix, we can rewrite the system in matrix form:
| -8 -8 | | x | | -16 |
| 6 -9 | * | y | = | -108 |
First, calculate the determinant of the coefficient matrix:
det( | -8 -8 |
| 6 -9 | ) = (-8)(-9) - (-8)(6) = -72 + 48 = -24
Since the determinant is not equal to zero, the system has a unique solution. To find the solution, we can use the inverse of the coefficient matrix:
| -8 -8 |^-1 = (1/-24) * | -9 8 |
| -6 -8 |
| -8 -8 |^-1 * | -16 | = (1/-24) * | -9 8 | * | -16 |
| -108 | | -6 -8 | | -108 |
= (1/-24) * | (-9)(-16) + (8)(-108) |
| (-6)(-16) + (-8)(-108) |
= (1/-24) * | 144 + (-864) |
| 96 + 864 |
= (1/-24) * | -720 |
| 960 |
= | -720/-24 |
| 960/-24 |
= | 30 |
| -40 |
So the solution to the system of equations is x = 30 and y = -40, which can be written as (30, -40).
Therefore, the correct answer is not provided in the options.
are you sure this is right?
I apologize for the mistake in my previous response. Let's find the correct solution using the matrix method.
The given system of equations can be written in matrix form as:
| -8 -8 | | x | | -16 |
| 6 -9 | * | y | = | -108 |
First, calculate the determinant of the coefficient matrix:
det( | -8 -8 |
| 6 -9 | ) = (-8)(-9) - (-8)(6) = 72 - (-48) = 72 + 48 = 120
Since the determinant is not equal to zero, the system has a unique solution. To find the solution, we can use the inverse of the coefficient matrix:
| -8 -8 |^-1 = (1/120) * | -9 8 |
| -6 -8 |
| -8 -8 |^-1 * | -16 | = (1/120) * | -9 8 | * | -16 |
| -108 | | -6 -8 | | -108 |
= (1/120) * | (-9)(-16) + (8)(-108) |
| (-6)(-16) + (-8)(-108) |
= (1/120) * | 144 + (-864) |
| 96 + 864 |
= (1/120) * | -720 |
| 960 |
= | -720/120 |
| 960/120 |
= | -6 |
| 8 |
So the solution to the system of equations is x = -6 and y = 8, which can be written as (-6, 8).
Therefore, the correct answer is A. (-6, 8).
To find the solution to the system of equations using a matrix, we can rewrite the equations in matrix form.
First, let's write down the coefficients of the variables as a matrix, called the coefficient matrix A:
A = [ -8 -8 ]
[ 6 -9 ]
Next, let's write down the constants on the right side of the equations as a column matrix, called the constant matrix B:
B = [ -16 ]
[ -108 ]
Now, we can form an augmented matrix by combining the coefficient matrix A and the constant matrix B:
[A | B] = [ -8 -8 | -16 ]
[ 6 -9 | -108 ]
To find the solution, we can perform row operations on the augmented matrix until we get it into reduced row-echelon form.
By performing the following row operations, we can reduce the augmented matrix:
1. Multiply the first row by 1/8.
[A | B] = [ 1 1 | 2 ]
[ 6 -9 | -108 ]
2. Multiply the first row by -6 and add it to the second row.
[A | B] = [ 1 1 | 2 ]
[ 0 -15 | -120 ]
3. Multiply the second row by -1/15.
[A | B] = [ 1 1 | 2 ]
[ 0 1 | 8 ]
4. Multiply the second row by -1 and add it to the first row.
[A | B] = [ 1 0 | -6 ]
[ 0 1 | 8 ]
From the reduced row-echelon form of the augmented matrix, we can read off the solution to the system of equations:
x = -6
y = 8
Therefore, the solution to the system of equations is (-6, 8), which corresponds to option A.