Use Cramer's rule to solve the following
3x-2y+5z=21
To use Cramer's rule to solve the equation 3x - 2y + 5z = 21, we need to set up a system of three equations with three variables:
Equation 1: 3x - 2y + 5z = 21
Equation 2: a1x + b1y + c1z = d1 (unknown)
Equation 3: a2x + b2y + c2z = d2 (unknown)
Next, we need to find the determinants of the coefficient matrix and the matrices obtained by replacing each column of the coefficient matrix with the constants on the right side, one column at a time.
The coefficient matrix is:
| 3 -2 5 |
| a1 b1 c1 |
| a2 b2 c2 |
Replacing the first column with the constants gives:
| 21 -2 5 |
| d1 b1 c1 |
| d2 b2 c2 |
Replacing the second column with the constants gives:
| 3 21 5 |
| a1 d1 c1 |
| a2 d2 c2 |
Replacing the third column with the constants gives:
| 3 -2 21 |
| a1 b1 d1 |
| a2 b2 d2 |
Now, we can calculate the determinant of the coefficient matrix using the formula:
| A | = a1(b2c3 - b3c2) - b1(a2c3 - a3c2) + c1(a2b3 - a3b2)
| A | = 3(b2c3 - b3c2) + 2(a2c3 - a3c2) - 5(a2b3 - a3b2)
Since only the first column contains the constants, the determinant of the coefficient matrix is simply:
| A | = 3
Next, we calculate the determinant of the matrix obtained by replacing the first column of the coefficient matrix with the constants:
| A1 | = 21(b2c3 - b3c2) + 2(d2c3 - d3c2) + 5(d2b3 - d3b2)
| A1 | = 21(b2c3 - b3c2) + 2(d2c3 - d3c2) + 5(d2b3 - d3b2)
Similarly, we calculate the determinants of the matrices obtained by replacing the second and third columns with the constants:
| A2 | = 3(a1c3 - a3c1) + 21(c1c3 - c2c1) + 5(a3c2 - a2c3)
| A3 | = 3(a1b2 - a2b1) + 2(a2c1 - a1c2) + 21(a1b3 - a3b1)
Now, we can solve for the variables using Cramer's rule:
x = | A1 | / | A |
y = | A2 | / | A |
z = | A3 | / | A |
Plugging in the determinants we calculated earlier:
x = | A1 | / | A | = (21(b2c3 - b3c2) + 2(d2c3 - d3c2) + 5(d2b3 - d3b2)) / 3
y = | A2 | / | A | = (3(a1c3 - a3c1) + 21(c1c3 - c2c1) + 5(a3c2 - a2c3)) / 3
z = | A3 | / | A | = (3(a1b2 - a2b1) + 2(a2c1 - a1c2) + 21(a1b3 - a3b1)) / 3
Note: Without the values of a1, b1, c1, a2, b2, c2, d1, and d2, we cannot calculate the specific values of x, y, and z. However, this is the general method using Cramer's rule to solve equations of this form.