Question

What are the different methods for solving a system of equations? Which method do you like the best? Why? When is your favorite method of solving a system of equations not the best method (please provide an example of that kind of system)?

Answer 1

The three methods used to solve systems of equation are substitution, elimination and augmented matrices. Substitution and elimination are simple methods that can effectively solve most systems of two equations in a few straightforward steps. The method of augmented matrices is little lengthier than other two methods.

Substitution
Substitution is the method of solving systems of equations by removing all but one of the variables in one of the equations and then solving that equation. This is achieved by isolating the other variable in an equation and then substituting values for these variables in other another equation. For example, to solve the system of equations x - y = 5, 2x - 3y = 3, isolate the variable x in the first equation to get x = 5+y, then substitute this value of x into the second equation to get 2(5 + y) - 3y = 3. This equation simplifies to -y = -7, or y = 7. substitute this value into the second equation to find the value of x: x - 7 = 5 or x = 12.

Elimination
Elimination is the method to solve systems of equations by rewriting one of the equations in terms of only one variable. The elimination method achieves this by adding or subtracting equations from each other in order to cancel out one of the variables. For example, adding the equations x + 2y = 5 and x - 2y = 7 yields a new equation, 2x = 12 or x = 6(note that the y terms cancelled out). The system is then solved using the same methods as for substitution. by substituting this value of x in first equation we get y =1/2. If it is impossible to cancel out the variables in the equations, it is necessary to multiply the entire equation by a factor to make the coefficients equal.

Augmented matrices
Augmented matrices can also be used to solve systems of equations. The augmented matrix consists of rows for each equation, columns for each variable, and an augmented column that contains the constant term on the other side of the equation. For example, the augmented matrix for the system of equations 2x + y = 4, 2x - y = 0 is [[2 1], [2 -1]...[4, 0]].

i like the method of elimination because it's easy and can be solved in less time tan the other two.

answer for the third part is in the attachment.