1. I will solve systems of linear equations using matrices and Gaussian elimination.
A multivariable (or multivariate) linear system is a system of linear equations in two or more variables.
Row-Echelon Form
Gaussian Elimination, named after the German mathematician Carl Friedrich Gauss, is the algorithm used to transform a system of linear equations into an equivalent system in row-echelon form.
Ex. Write the system of equations in triangular form using Gaussian elimination. Then solve the system.
x + 3y + 2z = 5
3x + y – 2z = 7
2x + 2y + 3z = 3
The augmented matrix of a system is derived from the coefficients and constant terms of the linear equations, each written in standard form with the constant terms to the right of the equal sign. If the column of constant terms is not included, the matrix reduces to that of the coefficient matrix of the system.
Ex. Write the augmented matrix for the system of linear equations.
x + y – z = 5
2w + 3x – z = –2
2w – x + y = 6
Compare Guassian elimination to matrix row operations
x + 3y + 2z = 5
3x + y – 2z = 7
2x + 2y + 3z = 3
Ex. Determine whether the following are in row-echelon form
a) b) c)
Ex. RESTAURANTS Three families ordered meals of hamburgers (HB), French fries (FF), and drinks (DR). The items they ordered and their total bills are shown below. Write and solve a system of equations to determine the cost of each item.
2. I will solve systems of linear equations using matrices and Gauss-Jordan elimination.
Gauss-Jordan elimination is the process of solving a system by transforming an augmented matrix so that it is in reduced row-echelon form.
Ex. Solve the system of equations.
x – y + z = 3
–x + 2y – z = 2
2x – 3y + 3z = 8
Ex. Solve the system of equations.
A)
x + 2y + z = 8
2x + 3y – z = 13
x + y – 2z = 5
B)
2x + 3y – z = 1
x + y – 2z = 5
x + 2y + z = 8
C)
4w + x + 2y – 3z = 10
3w + 4x + 2y + 8z = 3
w + 3x + 4y + 11z = 11