1. The problem statement, all variables and given/known data 2x +5y -z = 3 3x + 5y -4z = -6 the bracket needs to go infront, so x,y,z satisfy all the equations at once. 4x -2y +6z = 1 2. Relevant equations Cramer's method: (I don't know how to add certain symbols to the text so it is difficult to express it) If A were the matrix of the coefficients of the variables then aij is a corresponding element and Aij = (-1)i+j * Mij - the subdeterminant? Is it the correct word? Mij is a minor of the corresponding element. So Cramer's idea is this: |A| = a11 * A11 + a21 * A21 + a31 * A31 and a12 * A11 + a22 * A21 + a32 * A31 = 0 and a13 * A11 + a23 * A21 + a33 * A31 = 0 3. The attempt at a solution I have no problem to solve this system, but the problem is: I get the correct determinant value , but the following equations (which need to be 0) are not zero. Is Cramer's idea flawed? I am certain I have done the calculations correctly. |A| = 2 * 38 - 3*28 - 4*15 = -68 good 5*38 - 5*28 + 2*15 =/= 0 not good -38 + 4*28 - 6*15 =/= 0 definitely not good I can also skip to the part where I replace the x,y,z coefficients with the numbers on the right side of the equal sign when I solve for x,y,z , but I am not so much interested in the solution, am I even allowed to do that operation when the requirements are not met? I can also say for certain that there is a solution to it, because |A| =/= 0.