Cramer's Rule Calculator

Cramer's Rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It relies heavily on determinants. For a 2x2 system of equations in the form ax + by = e and cx + dy = f, the main determinant (W) is calculated as (a * d) - (b * c). The values for x and y are then found by dividing the specific variable determinant (Wx for x, Wy for y) by the main determinant W. For 3x3 systems, the process expands to compute 3x3 determinants using cofactor expansion or rule of Sarrus, calculating W, Wx, Wy, and Wz to solve for x, y, and z respectively.

Let us solve a 2x2 system of linear equations using Cramer's rule: 2x + 3y = 8 and 4x - y = 2. First, identify the coefficients: a1 = 2, b1 = 3, d1 = 8, a2 = 4, b2 = -1, and d2 = 2. Next, calculate the main determinant W = (2 * -1) - (3 * 4) = -2 - 12 = -14. Then, compute the determinant for x by replacing the x-coefficients with the constants: Wx = (8 * -1) - (3 * 2) = -8 - 6 = -14. Compute the determinant for y by replacing the y-coefficients: Wy = (2 * 2) - (8 * 4) = 4 - 32 = -28. Finally, divide to find the variables: x = Wx / W = -14 / -14 = 1, and y = Wy / W = -28 / -14 = 2. The final solution is x = 1 and y = 2.

Always double-check your coefficient entry to ensure negative signs are correctly applied, as a single sign error will corrupt all determinant values. Verify that the main determinant (W) is not equal to zero; if W equals zero, the system either has no solution or infinitely many solutions, meaning Cramer's rule cannot yield a unique answer. Utilize fraction form when dealing with non-integer determinants to maintain maximum mathematical precision throughout your assignment or analysis.