Solve a system of equations using augmented matrices

Learning Objectives Convert a system of linear equations into its corresponding augmented matrix. Identify and perform the three elementary row operations on a matrix. Transform an augmented matrix into row-echelon form using a systematic approach. Use back-substitution to find the unique solution from a matrix in row-echelon form. Solve systems of two and three linear equations using the augmented matrix method. Interpret the final matrix form to determine if a system has one solution, no solution, or infinitely many solutions. Ever tried to solve a puzzle with multiple clues? 🧩 What if you could organize those clues into a powerful grid to find the answer systematically? In this tutorial, you will learn how to solve systems of linear equations using a powerful tool calle...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that share the same variables.The equations 2x + y = 7 and x - 3y = -7 form a system of two equations with two variables, x and y. MatrixA rectangular grid or array of numbers arranged in rows and columns.A 2x3 matrix has 2 rows and 3 columns: [ [5, 0, -2], [1, 4, 9] ] Augmented MatrixA single matrix that represents a system of linear equations. It's formed by the coefficient matrix and the constant terms, separated by a vertical line.For the system 2x + y = 7 and x - 3y = -7, the augmented matrix is [ [2, 1 | 7], [1, -3 | -7] ]. Coefficient MatrixThe part of an augmented matrix that contains only the coefficients of the variables.For the system 2x + y = 7 and x - 3y = -7, the coefficient matrix is...

Elementary Row Operation 1: Row Swapping R_i \leftrightarrow R_j You can interchange any two rows of the matrix. This is equivalent to changing the order of the equations in the system. Elementary Row Operation 2: Row Multiplication kR_i \rightarrow R_i \quad (k \neq 0) You can multiply all the elements in a single row by any non-zero constant. This is equivalent to multiplying both sides of an equation by the same non-zero number. Elementary Row Operation 3: Row Addition R_i + kR_j \rightarrow R_i You can replace a row with the sum of that row and a non-zero multiple of another row. This is the most powerful operation, equivalent to adding a multiple of one equation to another.