use a matrix to find the solution of the system of equations
-8-9y=-16
6x-9y=-108
To solve the system of equations using matrices, we need to express the equations in matrix form. Let's start by organizing the coefficients of variables:
| -9y = -16 |
| 6x - 9y = -108 |
Next, let's rearrange the equations to isolate the variables:
| -9y = -16 |
| 6x - 9y = -108 |
Dividing the first equation by -9, we have:
| y = 16/9 |
| 6x - 9y = -108 |
Now, we can substitute the value of y in the second equation to solve for x:
| y = 16/9 |
| 6x - 9(16/9) = -108 |
Further simplifying the equation, we obtain:
| y = 16/9 |
| 6x - 16 = -108 |
Rearranging the equation, we have:
| y = 16/9 |
| 6x = -108 + 16 |
Simplifying the equation, we get:
| y = 16/9 |
| 6x = -92 |
Dividing both sides of the equation by 6, we find:
| y = 16/9 |
| x = (-92)/6 |
Reducing the fraction, the solution is:
| y = 16/9 |
| x = -46/3 |
Therefore, the solution to the system of equations is x = -46/3 and y = 16/9.
The given system of equations can be rewritten in matrix form as:
[-8, -9] [x] = [-16]
[ 6, -9] [y] [-108]
Let's denote the matrix on the left-hand side as A, the column vector on the right-hand side as B, and the column vector containing the variables x and y as X. Then, we have the equation AX = B.
To find X, we need to solve for X in the equation AX = B. We can do this by multiplying both sides by the inverse of matrix A. So, we have X = A^(-1) * B.
First, let's calculate the inverse of matrix A:
A = [-8, -9]
[ 6, -9]
The inverse of a 2x2 matrix can be calculated as:
A^(-1) = (1 / det(A)) * [ -9, 9]
[-6, -8]
where det(A) denotes the determinant of matrix A.
The determinant of matrix A can be found as (-8 * -9) - (6 * -9) = 72 - (-54) = 72 + 54 = 126.
So, the inverse of matrix A is:
A^(-1) = (1 / 126) * [ -9, 9]
[ -6, -8]
Now, let's calculate the product A^(-1) * B:
A^(-1) * B = (1 / 126) * [ -9, 9] * [ -16 ]
[ -6, -8] [ -108]
Multiplying these matrices together, we get:
A^(-1) * B = (1 / 126) * [ (-9 * -16) + (9 * -108) ]
[ (-6 * -16) + (-8 * -108) ]
Simplifying further:
A^(-1) * B = (1 / 126) * [ 144 + (-972) ]
[ 96 + 864 ]
A^(-1) * B = (1 / 126) * [ -828 ]
[ 960 ]
Finally, let's multiply the scalar (1 / 126) by each element in the resulting matrix:
A^(-1) * B = [ -828 / 126 ]
[ 960 / 126 ]
Simplifying further:
A^(-1) * B = [ -6.5714 ]
[ 7.619 ]
So, the solution to the system of equations is x = -6.5714 and y = 7.619.
To use a matrix to find the solution of a system of equations, we need to represent the coefficients of the variables and the constants in matrix form. Let's call the coefficient matrix A, the variable matrix X, and the constant matrix B.
The given system of equations is:
-8 - 9y = -16 (equation 1)
6x - 9y = -108 (equation 2)
We can rewrite the system of equations in matrix form as:
AX = B
Here, A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
To form the matrix A, we need to take the coefficients of the variables:
A = [ 0 -9 ]
[ 6 -9 ]
Next, we form the matrix X, which represents the variables:
X = [ x ]
[ y ]
Finally, we create the matrix B, which holds the constant values:
B = [ -8 ]
[ -108 ]
So, the system of equations can be represented as:
[ 0 -9 ] [ x ] = [ -8 ]
[ 6 -9 ] [ y ] [ -108 ]
To find the solution, we can use matrix inversion. We'll multiply both sides of the equation by the inverse of matrix A:
A^-1 * A * X = A^-1 * B
Since A^-1 * A results in the identity matrix (I), and I * X = X, we have:
X = A^-1 * B
To find the inverse of matrix A, we can use various methods like the Gauss-Jordan elimination algorithm or matrix calculator software.
Once we have the inverse of A, we can multiply it by B to get the variable matrix X, which will give us the solution to the system of equations.