How can you represent the system of equations witha matrix?
11x - 12y = 7
5x + 12y = -7
To represent the system of equations with a matrix, we need to create a coefficient matrix, variable matrix, and constant matrix.
The coefficient matrix contains the coefficients of the variables in each equation. In this case, the coefficient matrix is:
| 11 -12 |
| 5 12 |
The variable matrix contains the variables, in this case, it will be:
| x |
| y |
The constant matrix contains the constants on the right side of each equation. In this case, it will be:
| 7 |
| -7 |
Combining these three matrices, we get:
| 11 -12 | | x | | 7 |
| 5 12 | | y | = | -7 |
To represent the system of equations with a matrix, we can create a coefficient matrix and a constant matrix.
The coefficient matrix, or [A], is formed by placing the coefficients of the variables in each equation as rows. In this case, the coefficient matrix will be:
[11 -12]
[5 12]
The constant matrix, or [B], is formed by placing the constants from the right side of each equation as rows. In this case, the constant matrix will be:
[7]
[-7]
We can then combine [A] and [B] to form an augmented matrix, [AB], which represents the system of equations:
[11 -12 | 7]
[5 12 |-7]
To represent a system of equations using a matrix, you can follow these steps:
1. Write down the coefficients of the variables in each equation. In this case, we have:
Equation 1: 11x - 12y = 7
Equation 2: 5x + 12y = -7
2. Create a matrix with the coefficients. The matrix should have one row for each equation and one column for each variable. For example, for the given system of equations, we have:
| 11 -12 | | x | | 7 |
| 5 12 | * | y | = | -7 |
3. Simplify the matrix multiplication. Multiply the coefficients matrix by the variables matrix.
| 11x - 12y | | 7 |
| 5x + 12y | = | -7 |
4. This represents the system of equations in matrix form. The left-hand side of the equation corresponds to the variables multiplied by their coefficients, and the right-hand side represents the constants of the equations.
So, the system of equations can be represented as:
| 11 -12 | | x | | 7 |
| 5 12 | * | y | = | -7 |