Use Cramer's Rule to solve: 3x-5y=21 and 4x+2y=2.
a=3, b=-5, c=4, d=2, e=21, f=2
21(2)-(-5)(2) divided by 3(2)-(-5)(4)
=52/26= x=2
3(2)-21(4) divided by 3(2)-(-5)(4)
=78/26= y=3
answer:
(2,3)
If you will "plug in" x = 2 and y=3 to either equation, you will see that your solution is incorrect.
Cramer's rule involves using the determinants of matrices. You do have the denominator determinant (26) correct. The numerator for y is wrong. It should be -78. y = -3
To use Cramer's Rule to solve a system of linear equations, follow these steps:
1. Write the equations in the form of Ax + By = C, where A, B, and C are coefficients and x and y are variables. For example, the given equations can be rewritten as 3x - 5y = 21 and 4x + 2y = 2.
2. Identify the coefficients of the variables and the constants. In this case, for the first equation, a = 3, b = -5, and c = 21. For the second equation, a = 4, b = 2, and c = 2.
3. Calculate the determinants of the coefficient matrix, the x matrix, and the y matrix. The coefficient matrix is formed by taking the coefficients of x and y from both equations and arranging them in a 2x2 matrix: |a b|, where a = 3, b = -5, c = 4, and d = 2. The x matrix is formed by replacing the coefficients of x with the constants from the equations: |c b|, where c = 21 and d = 2. The y matrix is formed by replacing the coefficients of y with the constants from the equations: |a c|, where a = 3 and c = 21.
4. Calculate the determinant of the coefficient matrix, also called the main determinant (D), using the formula D = (a*d) - (b*c), where a = 3, b = -5, c = 4, and d = 2. In this case, D = (3*2) - (-5*4) = 6 + 20 = 26.
5. Calculate the determinant of the x matrix (Dx) using the formula Dx = (c*d) - (b*d), where c = 21 and d = 2. In this case, Dx = (21*2) - (-5*4) = 42 + 20 = 62.
6. Calculate the determinant of the y matrix (Dy) using the formula Dy = (a*d) - (a*c), where a = 3 and c = 21. In this case, Dy = (3*2) - (3*21) = 6 - 63 = -57.
7. Finally, calculate the values of x and y using the formulas x = Dx/D and y = Dy/D. In this case, x = 62/26 = 2 and y = -57/26 = -3.
Therefore, the solution to the system of equations 3x - 5y = 21 and 4x + 2y = 2 is x = 2 and y = -3.