Problem 13 - College Algebra

Question

Write the linear system from the augmented matrix. \(\left[\begin{array}{rrr|r}3 & 2 & 0 & 3 \\ -1 & -9 & 4 & -1 \\ 8 & 5 & 7 & 8\end{array}\right]\)

Step-by-Step Solution

Answer

3x_1 + 2x_2 = 3, -x_1 - 9x_2 + 4x_3 = -1, 8x_1 + 5x_2 + 7x_3 = 8.

Step 1: Understand the Matrix Format

Identify the given augmented matrix \[\left[\begin{array}{rrr|r} 3 & 2 & 0 & 3 \ -1 & -9 & 4 & -1 \ 8 & 5 & 7 & 8 \end{array}\right]\]This represents a system of three equations. Each row corresponds to an equation, and each column (before the vertical bar) corresponds to coefficients of the variables.

Step 2: Assign Variables

Let's denote the variables as follows:- Let \(x_1\) correspond to the first column of coefficients.- Let \(x_2\) correspond to the second column of coefficients.- Let \(x_3\) correspond to the third column of coefficients.The column after the vertical bar gives the constants on the right-hand side of each equation.

Step 3: Write Equations from Rows

For each row of the matrix, write an equation using the coefficients and the constants.**First row:**The equation is:\[ 3x_1 + 2x_2 + 0x_3 = 3 \]**Second row:**The equation is:\[ -1x_1 - 9x_2 + 4x_3 = -1 \]**Third row:**The equation is:\[ 8x_1 + 5x_2 + 7x_3 = 8 \]

Step 4: Present the Linear System

The linear system of equations corresponding to the augmented matrix is:\[\begin{align*}3x_1 + 2x_2 &= 3 \-x_1 - 9x_2 + 4x_3 &= -1 \8x_1 + 5x_2 + 7x_3 &= 8\end{align*}\]

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool used in linear algebra to represent systems of linear equations. It is essentially a compact way of writing down the coefficients and constants of these equations in a matrix form.
In an augmented matrix, each row corresponds to an equation in the system while each column represents the coefficients of a specific variable. The last column, separated by a vertical line, contains the constants from the equations’ right-hand sides.
For example, the augmented matrix \[\left[\begin{array}{rrr|r} 3 & 2 & 0 & 3 \ -1 & -9 & 4 & -1 \ 8 & 5 & 7 & 8\end{array}\right]\] corresponds to a system of three equations with three variables.

Linear Equations

Linear equations are mathematical statements of equality involving linear expressions. They form straight lines when graphed on a coordinate plane and contain no exponents or powers greater than one.
A system of linear equations involves multiple such equations, which can be solved simultaneously to find the values of the variables involved.
From the augmented matrix, you can express the system of linear equations as follows:

• The first row translates to \(3x_1 + 2x_2 + 0x_3 = 3\).
• The second row translates to \(-1x_1 - 9x_2 + 4x_3 = -1\).
• The third row translates to \(8x_1 + 5x_2 + 7x_3 = 8\).

These equations are linear because each term is either a constant or a constant times a variable.

Variables and Coefficients

In the context of linear equations, variables are the unknowns that we aim to solve for, like \(x_1\), \(x_2\), and \(x_3\) in our example. Coefficients are the numbers that multiply the variables in each term of the equation.
In our augmented matrix, variables are assigned to specific columns:

• \(x_1\) corresponds to the first column of coefficients.
• \(x_2\) corresponds to the second column of coefficients.
• \(x_3\) corresponds to the third column of coefficients.

Learning to identify the variables and their respective coefficients is essential in setting up the correct equations from an augmented matrix, helping solve the system effectively.