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1. Substitution: Substitute one equation into another to solve for a variable. 2. Elimination: Add or subtract the equations to  eliminate one of the variables. 3. Gaussian elimination: Use row operations to convert the augmented matrix into reduced row-echelon form. 4. Cramer's rule: Express the solution in terms of determinants.Matrix inversion: Express the system as a matrix equation and find the inverse of the matrix.  Here are a few examples of how to solve simultaneous equations using different methods: Substitution: Example: Solve for x and y in the system of equations: y = x + 2 y = -x + 6 Solution: Substitute the first equation into the second: -x + 6 = x + 2 Solve for x: -2x = 4 x = -2 Substitute x = -2 back into the first equation to find y: y = -2 + 2 = 0 So the solution is x = -2, y = 0. Elimination: Example: Solve for x and y in the system of equations: 2x + y = 8  ---------(1)                                        x + 2y = 6 ----------(2) Solution: Multiply the firs