Solving Systems of Equations with Augmented Matrices

This lesson demonstrates how to solve a system of two equations with two unknowns using an augmented matrix. It explains how to write the equations in standard form, form the augmented matrix, and use the Desmos matrix calculator to convert it into reduced row echelon form. The solution is derived and verified, showing that the system is consistent and independent.

What is the purpose of using an augmented matrix in solving systems of equations?

• To simplify the equations

• To eliminate variables

• To find the determinant

• To represent the system in a matrix form

Why is it important to have equations in standard form before creating an augmented matrix?

• To ensure the matrix is square

• To align coefficients and constants correctly

• To make calculations easier

• To reduce the number of variables

What does the first row of the augmented matrix represent for the equation 3x + 2y = 7?

• 1 0 3

• 3 0 7

• 3 2 7

• 0 1 2

How do you enter the dimensions of a matrix in the Desmos matrix calculator?

• By using a slider

• By typing them directly

• By selecting from a dropdown

• By pressing plus to adjust columns

What is the reduced row echelon form of the matrix with rows [3, 2, 7] and [1, -2, -3]?

• [0, 1, 2] and [1, 0, 1]

• [1, 0, 1] and [0, 1, 2]

• [1, 1, 0] and [0, 0, 1]

• [1, 0, 0] and [0, 1, 1]

What does the ordered pair (1, 2) represent in the context of the solution?

• x = 0 and y = 1

• x = 1 and y = 0

• x = 2 and y = 1

• x = 1 and y = 2

What is the significance of having a diagonal of ones in the reduced row echelon form?

• It simplifies finding the solution

• It shows the system has no solution

• It indicates the matrix is singular

• It represents the identity matrix

How can you verify that the ordered pair (1, 2) satisfies the first equation 3x + 2y = 7?

• By graphing the equation

• By solving for y

• By substituting x = 1 and y = 2 into the equation

• By checking if 3 + 2 equals 7

What is the result of substituting x = 1 and y = 2 into the second equation x - 2y = -3?

• 1 - 4 = -3

• 1 + 4 = 5

• 1 - 2 = -1

• 1 + 2 = 3

What does it mean if a system of equations is classified as consistent and independent?

• It has no solution

• It has dependent equations

• It has infinitely many solutions

• It has exactly one solution