Use Cramer's rule to solve this:
3x+2y= 7 ----- (1)
5x-3y= 37 ----- (2)
To use Cramer's rule, we need to first find the determinant of the coefficient matrix and the determinant of the matrix formed by replacing the first column of the coefficient matrix with the constants from the right-hand side of the equations.
Coefficient matrix:
|3 2|
|5 -3|
Constants matrix (replacing first column):
|7 2|
|37 -3|
Determinant of coefficient matrix:
|3 2|
|5 -3| = (3)(-3) - (2)(5) = -9 - 10 = -19
Determinant of constants matrix (replacing first column):
|7 2|
|37 -3| = (7)(-3) - (37)(2) = -21 - 74 = -95
Now we can use Cramer's rule to solve for x and y:
x = (d_x/d)
where d_x is the determinant of the matrix formed by replacing the first column of the coefficient matrix with the constants from the right-hand side of the equations, and d is the determinant of the coefficient matrix.
y = (d_y/d)
where d_y is the determinant of the matrix formed by replacing the second column of the coefficient matrix with the constants from the right-hand side of the equations, and d is the determinant of the coefficient matrix.
So, plugging in the determinants we found earlier, we get:
x = (d_x/d) = (-95)/(-19) = 5
y = (d_y/d) = (-29)/(-19) = 1.53 (rounded to 2 decimal places)
Therefore, the solution to the system of equations is x = 5 and y = 1.53.
AAAaannndd the bot gets it wrong yet again!
|3 7|
|5 37| = 76
so y = 76/-19 = -4
I apologize for my mistake in the previous response. You are correct, the determinant of the coefficient matrix is indeed -19. Therefore, the correct solution using Cramer's rule is:
x = (d_x/d) = (-95)/(-19) = 5
y = (d_y/d) = (76)/(-19) = -4
Thank you for correcting me and I apologize for any confusion caused.
To solve the system of equations using Cramer's rule, we need to find the values of x and y by solving for the determinants of the system.
Step 1: Write the system of equations in matrix form:
| 3 2 | | x | | 7 |
| | x | | = |
| 5 -3 | | y | | 37 |
Step 2: Calculate the determinant of the coefficient matrix, D:
D = | 3 2 |
| 5 -3 |
D = (3 * -3) - (2 * 5)
= -9 - 10
= -19
Step 3: Replace the coefficients of x with the constants on the right side of the equation:
| 7 2 | | x | | 7 |
| | x | | = |
| 37 -3 | | y | | 37 |
Step 4: Calculate the determinant of Dx, where Dx is formed by replacing the coefficients of x:
Dx = | 7 2 |
| 37 -3 |
Dx = (7 * -3) - (2 * 37)
= -21 - 74
= -95
Step 5: Replace the coefficients of y with the constants on the right side of the equation:
| 3 7 | | x | | 7 |
| | x | | = |
| 5 37 | | y | | 37 |
Step 6: Calculate the determinant of Dy, where Dy is formed by replacing the coefficients of y:
Dy = | 3 7 |
| 5 37 |
Dy = (3 * 37) - (7 * 5)
= 111 - 35
= 76
Step 7: Finally, calculate the values of x and y using Cramer's rule:
x = Dx / D
= -95 / -19
= 5
y = Dy / D
= 76 / -19
= -4
Therefore, the solution to the system of equations is x = 5 and y = -4.