How to solve simultaneous equation

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 first equation by 2: 4x + 2y = 16 

Subtract the second equation from the first:

 3x = 10 Solve for x: x = 10/3

 Substitute x = 10/3 back into the second equation to find y: 10/3 + 2y = 6 

Solve for y: y = (6 - 10/3) / 2 = (6 - 10)/6 = -2/3 

So the solution is x = 10/3, y = -2/3.