How to solve an equation with three unknowns

If two of the three equations of the system have only two unknown of the three, try to Express some variables through others and substitute them in the equation with three unknown. Your goal is to turn it into a normal equation with one unknown. If this is successful, then the solution is quite simple – substitute the value found into the other equation and find all the other unknown.

Some systems of equations can be solved by subtracting one equation from the other. Let's see if there are opportunities to multiply one expression by the number or variable so that when subtraction was reduced from two unknown. If so, use it, most likely, a subsequent decision is not difficult. Don't forget that when multiplying by a number, multiply both the left and right. Similarly, when you subtract the equations you need to remember that the right side must also be deducted.

If the previous methods did not help, use the General method of solutions of any equations with three unknowns. To do this, rewrite the equation in the form а11х1+а12х2+а13х3=b1, а21х1+а22х2+а23х3=b2, а31х1+а32х2+а33х3=b3. Now make a matrix of the coefficients x (A), the unknown matrix (X) matrix and free members (In). Please note, multiplying the coefficient matrix by a matrix is unknown, you will get a matrix equal to the matrix free members, that is, A*X=B.

Find the matrix a of degree (-1) after finding the determinant of a matrix, note that it should not be zero. Then, multiply the resulting matrix on the matrix, as a result you get the desired matrix X, showing all the values.

Find the solution of the system of three equations using the method of Kramer. To do this, find the determinant of the third order ∆, the corresponding matrix system. Then find three more of the determinant ∆1, ∆2 and ∆3, by substituting values of the corresponding columns of the values of the free members. Now find x: x1=∆1/∆, x2=∆2/∆, X3=∆3/∆.