Problem 27 - Elementary and Intermediate Algebra

Question

Explain how you can recognize dependent equations when solving with matrices.

Step-by-Step Solution

Answer

Dependent equations are identified when a row in the augmented matrix becomes all zeros after row operations.

Step 1: Identify the System of Equations

Begin by writing down the system of linear equations you need to solve. For example, let's use:\[\begin{align*}2x + 3y &= 6 \4x + 6y &= 12\end{align*}\]

Step 2: Formulate the Augmented Matrix

Convert the system of equations into an augmented matrix form:\[\left[ \begin{array}{cc|c}2 & 3 & 6 \4 & 6 & 12\end{array} \right]\]

Step 3: Perform Row Operations

Use row operations to simplify the matrix. For this example, you can subtract 2 times the first row from the second row, giving:\[\left[ \begin{array}{cc|c}2 & 3 & 6 \0 & 0 & 0\end{array} \right]\]

Step 4: Identify Dependent Equations

Check the simplified matrix. If any row has all zeros on the left-hand side and the right-hand side (as in our example with the second row being all zeros), this means the system has dependent equations.

Step 5: Conclusion Based on Matrix Form

Since the second row has all zeros, it indicates the second equation does not provide new information, which means it is dependent on the first equation. Thus, the original set of equations are dependent.

Key Concepts

System of Linear Equations

A system of linear equations comprises multiple linear equations that share the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. For instance, consider the following system:

\[\begin{align*}2x + 3y &= 6 4x + 6y &= 12\end{align*}\]In this case, the variables are \(x\) and \(y\). We want to find out if there are any values for \(x\) and \(y\) that can satisfy both equations at once.

Augmented Matrix

An augmented matrix is a simplified way to represent a system of linear equations. It merges the coefficients from each equation and the constants into a single matrix.

For the system above, the augmented matrix looks like this:

\[\left[ \begin{array}{cc|c}2 & 3 & 6 4 & 6 & 12\end{array} \right]\]Each row of the matrix corresponds to one equation, and each column aligns with a variable, except for the final column, which represents constants. Using an augmented matrix simplifies row operations, which are key to identifying dependent or independent equations.

Row Operations

Row operations are techniques used to manipulate matrices into simpler forms, helping solve systems of linear equations more efficiently. Three main types of row operations include:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting the multiple of one row to another row

In our case, let's perform a row operation on the augmented matrix by subtracting 2 times the first row from the second row:

\[\left[ \begin{array}{cc|c}2 & 3 & 6 0 & 0 & 0\end{array} \right]\]Our goal is to create zeros in strategic places to make the system easier to solve. Here, we can see the second row has turned into all zeros.

Linear Dependence

Linear dependence occurs when one equation in a system can be derived from another by multiplying by a constant or adding/subtracting multiple equations. This means the dependent equation does not provide any new information.

In the context of our example, after row operations, we have:

\[\left[ \begin{array}{cc|c}2 & 3 & 6 0 & 0 & 0\end{array} \right]\]The second row of all zeros indicates the second equation is completely dependent on the first. Checking dependent equations is crucial to understand the nature and solutions of the system. If a row consists entirely of zeros (both left-hand side and right-hand side), it is indicative of dependent equations.