Problem 19 - College Algebra with Modeling and Visualization

Question

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & 5 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution is \((3 - 3t, 2t + 1, t)\) for any real number \( t \).

Step 1: Understand the Matrix

Examine the given augmented matrix: \[\begin{array}{rrr|r} 1 & 2 & -1 & 5 \0 & 1 & -2 & 1 \0 & 0 & 0 & 0\end{array}\]The matrix provides coefficients for a system of linear equations. This matrix corresponds to the following system of equations: \[\begin{align*}1x + 2y - 1z &= 5, \0x + 1y - 2z &= 1.\end{align*}\] There is no third equation since the third row is all zeros.

Step 2: Identify the Variable Relations

From the second equation: \[ y - 2z = 1\]express \( y \) in terms of \( z \): \[ y = 2z + 1.\] This expresses \( y \) as a dependent variable on \( z \). \( z \) is a free variable, allowing it to take any scalar value.

Step 3: Substitute to Solve for x

Substitute \( y = 2z + 1 \) back into the first equation: \[x + 2(2z + 1) - z = 5\] Simplify it to solve for \( x \):\[x + 4z + 2 - z = 5 \x + 3z = 3 \x = 3 - 3z.\]\( x \) is now expressed in terms of \( z \), where \( z \) remains the free variable.

Step 4: Formulate the Solution

Since \( z \) is a free variable, let \( z = t \) where \( t \) is any real number. Then the solution as an ordered triple \((x, y, z)\) becomes:\[ (3 - 3t, 2t + 1, t) \] This represents the parametric solution of the given system in terms of the free parameter \( t \).

Key Concepts

Row-Echelon Form

Understanding the row-echelon form is crucial when solving systems of linear equations using matrix algebra. When a matrix is in row-echelon form, it means that:

• Each nonzero row has more leading zeros than the previous row.
• The leading coefficient (or pivot) of a nonzero row is always to the right of the leading coefficient of the row directly above it.
• Any rows consisting entirely of zeros are at the bottom of the matrix.

This form helps simplify the process of solving systems of equations, as seen in the augmented matrix provided:\[\begin{array}{rrr|r}1 & 2 & -1 & 5 \0 & 1 & -2 & 1 \0 & 0 & 0 & 0\end{array}\]Here, the system is simplified enough to be solved with backward substitution, allowing us to efficiently find the variables' relations.

Linear Systems

Linear systems consist of linear equations, where each equation represents a straight line in a geometric space. For instance, the augmented matrix corresponds to the linear system:\[\begin{align*}1x + 2y - 1z &= 5, \0x + 1y - 2z &= 1.\end{align*}\]In essence, each equation is a plane within a three-dimensional space. Solving the linear system means finding the intersection of these planes, which represents the values of the variables that satisfy all equations simultaneously. The solution can be unique, infinite, or nonexistent, depending on the equations provided.

Parametric Solution

A parametric solution employs parameters to express the solutions of a system of equations. This approach is especially beneficial for systems that do not have a unique solution or when free variables are involved. In this case, the system has one free variable, \( z \), due to the absence of constraints from the third row of zeros in the matrix. Thus, \( z \) can be any real number, denoted by the parameter \( t \).Expressing other variables in terms of \( t \), we find:

• \( y = 2t + 1 \)
• \( x = 3 - 3t \)

This leads to the parametric solution \( (x, y, z) = (3 - 3t, 2t + 1, t) \), which describes a line of solutions governed by the variable \( t \).

Augmented Matrix

An augmented matrix presents a compact way to handle linear equations. It includes the coefficients and constants of multiple linear equations within a single matrix format. In the augmented matrix:\[\begin{array}{rrr|r}1 & 2 & -1 & 5 \0 & 1 & -2 & 1 \0 & 0 & 0 & 0\end{array}\]\[-\]- The part left of the vertical bar represents the coefficients of the variables \( x, y, \) and \( z \).- The portion to the right denotes the constants from each equation.Using the augmented matrix streamlines the process of applying row operations, rearranging equations, and identifying solution types. It facilitates solving systems by allowing easy conversion and manipulation, especially when transforming matrices to their row-echelon form to reveal linear relationships directly.