Problem 28 - A Graphical Approach to College Algebra

Question

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & -1 & 2 & 8 \\ 0 & 1 & -4 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The solution is \((x, y, z) = (2z + 10, 4z + 2, z)\).

Step 1: Understand the Matrix

The given augmented matrix represents a system of linear equations. Each row corresponds to an equation in the system. The matrix is in row echelon form, where the last row indicates that there is no equation for the variable associated with the third column. We can interpret these equations before solving: 1. \( x - y + 2z = 8 \) 2. \( y - 4z = 2 \) 3. The last row \( 0 = 0 \) is true and does not affect the solution.

Step 2: Solve for y from Row 2

From the second row \( y - 4z = 2 \), we solve for \( y \): \[ y = 4z + 2 \] This expression gives us \( y \) in terms of \( z \), which indicates that \( y \) depends on \( z \).

Step 3: Solve for x from Row 1

Using the first row \( x - y + 2z = 8 \), substitute \( y = 4z + 2 \):\[ x - (4z + 2) + 2z = 8 \]Simplify this equation to find \( x \):\[ x - 4z - 2 + 2z = 8 \]\[ x - 2z = 10 \]Thus, \( x = 2z + 10 \).

Step 4: Interpret the Solution

The variables \( x \) and \( y \) are expressed in terms of the parameter \( z \). Thus, the solutions are:- \( x = 2z + 10 \)- \( y = 4z + 2 \)- \( z = z \) (\( z \) is a free variable). This means there are infinitely many solutions dependent on the value of \( z \).

Key Concepts

Row Echelon Form

A matrix is in row echelon form when it satisfies certain conditions, making it easier to solve systems of linear equations using techniques like back-substitution. In this form, each leading entry (the first non-zero number from the left in a row) is to the right of any leading entries in rows above it.
This structure brings clarity, as it organizes the equations to show dependencies between variables. In our example, the matrix is already in row echelon form.

• The first row translates to the equation: \( x - y + 2z = 8 \).
• The second row reads as: \( y - 4z = 2 \).
• The third row, \( 0 = 0 \), indicates a redundant equation.

Working in this form helps isolate variables one by one, starting from the bottom and moving upwards.

Augmented Matrix

An augmented matrix is a useful tool for representing a system of linear equations. It combines the coefficients of variables and the constants from the equations' right side into a compact matrix form. This visual consolidation allows us to apply matrix operations more efficiently, leading to solving equations faster.
For the given system, the augmented matrix is:\[\begin{bmatrix}1 & -1 & 2 & | & 8 \0 & 1 & -4 & | & 2 \0 & 0 & 0 & | & 0\end{bmatrix}\]
The vertical line separates the coefficients from the solutions. Each row corresponds to one equation in the system, offering a clear overview of the problem at hand. Using such representations aids in applying methods like Gaussian elimination to reach row echelon form and solve efficiently.

Linear System Solutions

Solving linear systems represented by matrices like the one given involves expressing variables in terms of others when one or more free variables are present. Back-substitution is used when a matrix is in row echelon form, allowing us to find possible solutions step by step.

• The first equation, \( x - y + 2z = 8 \), after substituting \( y = 4z + 2 \), becomes \( x - 2z = 10 \). Hence, \( x = 2z + 10 \).
• For \( y \), using the second row equation, we find \( y = 4z + 2 \).
• Since there is no equation in the third row, \( z \) is a free variable.

Such solutions are parametrized, meaning there are infinitely many solutions depending on the value of the free variable \( z \). Understanding this flexibility is key in linear algebra, showcasing that different values for \( z \) lead to various solutions for \( x \) and \( y \). This concept is central in understanding spaces and dimensions in linear algebra.