Problem 8 - Beginning and Intermediate Algebra

Question

Write a system of linear equations in \(x\) and \(y\) represented by each augmented matrix. $$\left[\begin{array}{ll|r}1 & 2 & 11 \\\0 & 1 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The system of linear equations represented by the given augmented matrix is: $$ \begin{cases} x + 2y = 11 \\ y = 3 \end{cases} $$

Step 1: Identify coefficients and constants

Using the given augmented matrix: $$\left[\begin{array}{ll|r}1 & 2 & 11 \\\0 & 1 & 3\end{array}\right]$$ We can see that the first row represents the coefficients and constants in the first equation: Coefficients: 1 (for x), 2 (for y) Constant: 11 The second row represents the coefficients and constants in the second equation: Coefficients: 0 (for x), 1 (for y) Constant: 3

Step 2: Write the linear equations

Now that we have the coefficients and constants for each equation, we can write the system of linear equations. First equation: \(1 \cdot x + 2 \cdot y = 11\) or simply, \(x + 2y = 11\) Second equation: \(0 \cdot x + 1 \cdot y = 3\) or simply, \(0x + y = 3\) or \(y = 3\) So the system of linear equations represented by the given augmented matrix is: $$ \begin{cases} x + 2y = 11 \\ y = 3 \end{cases} $$

Key Concepts

Augmented Matrix

An augmented matrix is an incredibly useful tool in solving and representing systems of linear equations. It takes the form of a rectangular array of numbers, which includes the coefficients of the variables and the constants from the equations. This compact representation helps simplify complex algebraic manipulations, making it easier to solve systems of equations.

To build an augmented matrix:

• Write down the coefficients of each variable from the given system of equations.
• Append the same row with the resulting constant from each equation, separated by a vertical line.

The augmented matrix ties each set of coefficients to its corresponding constant, allowing operations such as row reduction to find solutions efficiently. By working with an augmented matrix, you can visualize and easily manipulate the equations in matrix form, streamlining the solving process.

Linear Equations

Linear equations are algebraic expressions that form straight lines when graphed on a coordinate plane. Each linear equation represents a line, and when you have multiple linear equations, you define a system.

In the helpful step-by-step solution provided, the linear equations, based on the matrix, were derived as:

• First equation: \( x + 2y = 11 \)
• Second equation: \( y = 3 \)

These equations are simple yet powerful, revealing the precise relationships between the variables.

In the system's context:

• The first equation involves both \( x \) and \( y \), showing a direct correlation.
• The second equation is simplified as it only involves \( y \), resulting in a straight horizontal line on a graph.

Understanding how these equations interact forms the foundation for solving systems using methods like substitution or elimination.

Algebra

Algebra is the branch of mathematics concerning variables and the rules for manipulating these symbols. It is the vital language of the system of linear equations. Algebra helps in expressing relationships and finding unknowns.

When dealing with systems using matrices, algebraic techniques such as simple matrix row operations are employed to simplify and solve the equations. These operations can include:

• Swapping rows.
• Multiplying a row by a scalar.
• Adding or subtracting rows.

These methods allow you to work towards finding the values of unknown variables. In the context of the original solution, algebra was used to translate the augmented matrix into clearly defined linear equations.

By understanding and applying algebra, you can effectively approach and solve a wide variety of mathematical problems, translating real-world situations into solvable equations.