Use a matrix to find the solution to the system of equations.
{-8x-8y=-16
{6x-9y=-108
(–6, 8)
(–6, 8)
(6, 8)
(6, 8)
(8, –6)
(8, –6)
(6, –8)
(6, –8)
(6, 8)
(6, 8) is the solution to the system of equations.
To solve the system of equations using a matrix, we can set up an augmented matrix with the coefficients of the variables and the constants.
The augmented matrix for the given system of equations is:
[-8 -8 | -16]
[6 -9 | -108]
To solve this matrix, we will perform row operations to get it in row-echelon form:
R2 = R2 + (3/4)R1:
[-8 -8 | -16]
[0 -15 | -112]
Next, we will divide the second row by -15 to get a leading coefficient of 1:
R2 = (1/15)R2:
[-8 -8 | -16]
[0 1 | 7.47]
Now, we will perform row operations to eliminate the coefficient of x in the first row:
R1 = R1 - (-8)R2:
[-8 0 | 49.47]
[0 1 | 7.47]
Finally, we will divide the first row by -8 to get a leading coefficient of 1:
R1 = (1/8)R1:
[-1 0 | -6.18]
[0 1 | 7.47]
From this row-echelon form, we can read off the solution to the system of equations:
x = -6.18
y = 7.47
Therefore, the solution to the given system of equations is (–6.18, 7.47).
To solve the system of equations using a matrix, we can use the method of matrix inversion. Here's how to proceed:
Step 1: Write the system of equations in matrix form.
The given system of equations can be represented as the following augmented matrix:
|-8 -8 | -16 |
| 6 -9 | -108|
Step 2: Find the inverse of the coefficient matrix.
To find the inverse of the coefficient matrix, we can use matrix operations. Let's call the coefficient matrix A and the inverse matrix A^-1.
| a b |
| c d |
The inverse matrix A^-1 is given by:
A^-1 = (1 / determinant of A) * adjoint of A
Where the adjoint of A is denoted as adj A and is obtained by swapping the elements in the main diagonal and changing the signs of the other elements.
Step 3: Calculate the determinant of the coefficient matrix.
The determinant of a 2x2 matrix A can be calculated by applying the formula:
det(A) = (a * d) - (b * c)
In this case, the determinant of the coefficient matrix is:
det(A) = (-8 * -9) - (-8 * 6)
Step 4: Calculate the adjoint of the coefficient matrix.
To find the adjoint of a 2x2 matrix A, we swap the elements in the main diagonal and change the signs of the other elements.
The adjoint of the coefficient matrix is:
adj A = | -9 8 |
| -6 -8 |
Step 5: Calculate the inverse of the coefficient matrix.
Now, we can calculate the inverse matrix A^-1 using the formula mentioned earlier:
A^-1 = (1 / det(A)) * adj A
Step 6: Multiply the inverse matrix with the augmented matrix.
To find the solution to the system of equations, we multiply the inverse matrix A^-1 with the augmented matrix. This will give us a matrix of the form:
| x |
| y |
Step 7: Simplify the matrix to obtain the solution.
After multiplying the inverse matrix A^-1 with the augmented matrix, you will get a matrix in the form:
| x | | constant |
| y | = | constant |
The values of x and y will be your solution to the system of equations.
In this case, you can follow the steps above to calculate the inverse matrix and obtain the solution to the system of equations.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
The length of a rectangle is 8.75 cm more than 3 times the width. If the perimeter of the rectangle is 89.5 cm, what are its dimensions?
Let's assume the width of the rectangle is x cm.
According to the given information, the length of the rectangle is 8.75 cm more than 3 times the width.
So, the length of the rectangle would be (3x + 8.75) cm.
The formula for the perimeter of a rectangle is given by:
Perimeter = 2(length + width)
Substituting the length and width values into the equation, we get:
89.5 = 2((3x + 8.75) + x)
Simplifying the equation:
89.5 = 2(4x + 8.75)
89.5 = 8x + 17.5
Rearranging the equation:
8x = 89.5 - 17.5
8x = 72
Dividing both sides by 8:
x = 9
So, the width of the rectangle is 9 cm.
Substituting this value back into the length equation:
Length = 3x + 8.75
Length = 3 * 9 + 8.75
Length = 27 + 8.75
Length = 35.75
Therefore, the dimensions of the rectangle are:
Width = 9 cm
Length = 35.75 cm