Problem 2 Based on Consistency in Equation

TL;DR

Solving a system of linear equations using matrix form and row transformations to find a consistent and unique solution.

Transcript

hi everyone today we are going to discuss problem number two based on consistency in linear equations so let me start problem is given here solve the following system of linear equations equations as given to you x 1 plus x 2 plus twice x 3 is equal to 8 second equation minus x 1 minus twice x 2 plus thrice x 3 is equal to 1 and third equation thri... Read More

Key Insights

• ☺️ Linear equations can be represented in matrix form as A*x = b, where A is the coefficient matrix, x is the unknown matrix, and b is the matrix of constants.
• 🤨 Row transformations are used to reduce the augmented matrix to echelon form, simplifying the system of equations.
• 😜 The rank of the coefficient matrix and augmented matrix is compared to determine consistency and the number of solutions.
• 💁 The solution of a consistent system of equations is found by back-substituting the values from the echelon form of the augmented matrix.

Questions & Answers

Q: How do you convert a system of linear equations into matrix form?

To convert a system of linear equations into matrix form, you write it as A*x = b, where A is the coefficient matrix, x is the unknown matrix, and b is the matrix of constants.

Q: What is the purpose of row transformations in solving linear equations?

Row transformations are used to reduce the augmented matrix to echelon form, which simplifies the system of equations and makes it easier to find the solution.

Q: How do you determine if a system of linear equations is consistent?

If the rank of the coefficient matrix is equal to the rank of the augmented matrix, and both ranks are equal to the number of unknowns, then the system is consistent.

Q: How do you find the solution of a consistent system of linear equations?

Once the system is consistent, you can use the echelon form of the augmented matrix to find the values of the unknown variables. This is done by back-substituting the values starting from the bottom row.

Summary & Key Takeaways

• The video discusses how to convert a system of linear equations into matrix form.

• Row transformations are used to reduce the augmented matrix to its echelon form.

• The rank of the coefficient matrix and augmented matrix is compared to determine consistency and find the solution.