Solve System of Equations: Gaussian Elimination

In summary, to solve the system of equations using Gaussian elimination, you need to first swap row 1 and row 2, then multiply row 1 by -3 and add it to row 2. Next, add 3 times row 1 to row 3 and row 4. This should lead to a solution for w, x, y, and z. Don't worry about fractions and make sure your arithmetic is accurate.
  • #1
psychfan29
9
0
Solve the following system of equations using Gaussian elimination:
3w+10x-2y+3z=55
w+12x-11y-z= 69
-3w-6x-5y-10z=-47
-3w+9x+5y+4z=-1

Here's what I did so far:
swap R1 and R2 (row 1 and row 2)
-3R1+R2 (multiply row 1 by -3 and add row 2 to that product)
3R1+R3
3R1+R4

Now, after that I think I may have made a mistake because I eventually get some very odd fractions that could not possibly be correct. Can somebody tell me what the steps are?
 
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  • #2
I followed your initial steps and finishing the elimination works out just fine despite any fractions. Make sure your arithmetic is correct and don't be afraid of fractions.
 

1. What is Gaussian Elimination and how does it work?

Gaussian Elimination is a method used to solve systems of linear equations. It involves using a sequence of elementary row operations to transform the system into an equivalent system that is easier to solve. These operations include adding or subtracting rows, multiplying rows by a constant, and switching rows. The goal is to reduce the system to a triangular form, making it easier to find the solution.

2. When is Gaussian Elimination the best method to solve a system of equations?

Gaussian Elimination is best used when the system of equations has multiple variables and equations, as it can handle larger systems efficiently. It is also useful when working with real-world problems that involve linear relationships between multiple variables.

3. What are the advantages of using Gaussian Elimination over other methods?

Gaussian Elimination is a systematic and straightforward method that guarantees a solution if one exists. It also reduces the chances of making mistakes compared to other methods such as substitution or graphing. Additionally, it can be easily programmed and automated, making it useful for solving complex systems of equations in computer programs.

4. Are there any limitations to using Gaussian Elimination?

One limitation of Gaussian Elimination is that it can become computationally expensive for very large systems of equations. It also requires careful handling of fractions and decimals, which can increase the chance of errors. Additionally, this method may not work for systems with inconsistent or dependent equations.

5. Can Gaussian Elimination be used for non-linear systems of equations?

No, Gaussian Elimination can only be used to solve linear systems of equations. Non-linear systems require different methods, such as substitution or graphing, to find a solution.

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