write the following simultaneous equations in the form of AX= B where A,X and B are matrices
11x+6y=6
9x+5y=7
hence write the solution for x and y as a product of two matrices.
this is the 5th time am posting this in the last 3 weeks and no one has helped me! i really don't know how to answer this question
Hard to write matrices in this format
You will have to visualize the matrix brackets around the matrices . I also will have to write them vertically.
11 6
9 5
times
x
y
=
6
7
To represent the simultaneous equations in the form of AX = B, we need to write the coefficients of x and y in matrix form.
The given equations are:
11x + 6y = 6 ... (1)
9x + 5y = 7 ... (2)
Let's define the following matrices:
A = [11 6]
[9 5]
X = [x]
[y]
B = [6]
[7]
Now, we can write the simultaneous equations in matrix form as AX = B:
A * X = B
⇒ [11 6] * [x] = [6]
[9 5] [y] [7]
To find the solution for x and y, we need to find the inverse of matrix A and multiply it with matrix B:
A^{-1} * A * X = A^{-1} * B
Note: A^{-1} represents the inverse of matrix A.
To find the inverse of matrix A, we can use various methods like Gaussian elimination or matrix inverse formula. Let's calculate the inverse using the matrix inverse formula:
A^{-1} = (1 / determinant(A)) * adjoint(A)
The determinant of A can be calculated as:
det(A) = 11 * 5 - 9 * 6 = 55 - 54 = 1
Now, we can find the adjoint of matrix A:
adjoint(A) = [5 -6]
[-9 11]
Therefore, A^{-1} = (1 / 1) * [5 -6]
[-9 11]
= [5 -6]
[-9 11]
Now, we can find the value of X by multiplying A^{-1} with B:
X = A^{-1} * B
⇒ [5 -6] * [6] = [5(6) + (-6)(7)] = [30 - 42]
[-9 11] [7] = [-54 + 77]
Therefore, the solution for x and y in matrix form is:
X = [x] = [30 - 42]
[y] [-54 + 77]
Hence, the solution for x and y can be written as a product:
[x] = [30 - 42] = 30 - 42 = -12
[y] [-54 + 77] -54 + 77 23
Therefore, the solution for x = -12 and y = 23.
To write the given simultaneous equations in the form of AX = B, where A, X, and B are matrices, we need to separate the coefficients of variables x and y into matrices.
The system of equations can be represented as:
A * X = B
Where:
A = |11 6|
| 9 5|
X = | x |
| y |
B = | 6 |
| 7 |
Now, to find the solution for x and y as a product of two matrices, we can solve the equation AX = B by multiplying both sides by the inverse of matrix A.
A * X = B
A^(-1) * (A * X) = A^(-1) * B
(X = A^(-1) * B)
To calculate the inverse of matrix A, we can use the inverse formula:
A^(-1) = (1 / ((11 * 5) - (6 * 9))) * | 5 -6 |
| -9 11 |
Simplifying, we get:
A^(-1) = (1/1) * | 5 -6 |
| -9 11 |
A^(-1) = | 5 -6 |
| -9 11 |
Now, we can multiply the inverse of A with B to find X:
X = A^(-1) * B
| x | | 5 -6 | | 6 |
| y | = | -9 11 | * | 7 |
Simplifying further:
| x | | (5 * 6) + (-6 * 7) |
| y | = | (-9 * 6) + (11 * 7) |
| x | | 30 - 42 |
| y | = | -54 + 77 |
| x | | -12 |
| y | = | 23 |
Therefore, the solution for x and y in matrix form is:
| x | | -12 |
| y | = | 23 |