solve the equation using cramer's rule. 3x+2y=12 and 2x+3y=7
To solve this system of equations using Cramer's Rule, we first need to better understand the method.
Cramer's Rule is a method for solving systems of equations using determinants. For a system of n linear equations with n variables, it states that if the determinant of the coefficient matrix is non-zero, a unique solution exists, and the solution can be found by evaluating determinants of specific matrices.
Let's solve the given system of equations step by step using Cramer's Rule.
Step 1: Identify the coefficients and constants
The system of equations can be represented as:
Equation 1: 3x + 2y = 12
Equation 2: 2x + 3y = 7
The coefficients of the variables x and y can be represented by a matrix, called the coefficient matrix:
| 3 2 |
| 2 3 |
And the constants on the right-hand side of the equations can be represented as a column matrix:
| 12 |
| 7 |
Step 2: Calculate the determinant of the coefficient matrix (D)
The determinant of the coefficient matrix is denoted as D. In this case, D is given by:
D = | 3 2 |
| 2 3 |
Determinant of a 2x2 matrix is calculated as (ad - bc), thus:
D = (3 * 3) - (2 * 2)
D = 9 - 4
D = 5
Step 3: Calculate the determinant of the matrix with x-coefficients replaced by constants (Dx)
To find the value of x, we replace the x-coefficients in the coefficient matrix with the constants and then find the determinant of this modified matrix Dx.
| 12 2 |
| 7 3 |
Dx = (12 * 3) - (2 * 7)
Dx = 36 - 14
Dx = 22
Step 4: Calculate the determinant of the matrix with y-coefficients replaced by constants (Dy)
Similarly, to find the value of y, we replace the y-coefficients in the coefficient matrix with the constants and then find the determinant of this modified matrix Dy.
| 3 12 |
| 2 7 |
Dy = (3 * 7) - (12 * 2)
Dy = 21 - 24
Dy = -3
Step 5: Calculate the values of x and y
Now, we can calculate the values of x and y using the following formulas:
x = Dx / D
y = Dy / D
Substituting the calculated determinants:
x = 22 / 5
y = -3 / 5
So, the solution to the given system of equations is x = 22/5 and y = -3/5.
D = 5
Dx = 22
Dy = -3
So,
x=Dx/D = 22/5
y=Dy/D = -3/5