Problem 16 - Precalculus with Limits

Question

In Exercises 15-20, write the augmented matrix for the system of linear equations. \( \left\\{ \begin{array}{l} 7x + 4y = 22 \\ 5x - 9y = 15 \end{array} \right. \)

Step-by-Step Solution

Answer

The augmented matrix for the system of equations is \[ \begin{bmatrix} 7 & 4 & | & 22 \ 5 & -9 & | & 15 \end{bmatrix} \].

Step 1: Write the first equation as a row

Write the first equation \(7x + 4y = 22\) as a first row in the matrix. The coefficients of the equation (7 and 4) form the first two numbers in the row, and the third number is the constant on the right hand side of the equation (22). So, the first row is \[ [7, 4 | 22] \] in the augmented matrix.

Step 2: Write the second equation as a row

Write the second equation \(5x - 9y = 15\) as the second row in the matrix. The coefficients of the equation (5 and -9) form the first two numbers in the row, and the last number is the constant on the right hand side of the equation (15). So, the second row is \[ [5, -9 | 15] \] in the augmented matrix.

Step 3: Combine the rows

Finally, combine the rows from steps 1 and 2 to form the augmented matrix. The first row of the matrix is from the first equation and the second row of the matrix is from the second equation. Therefore, the final matrix is \[ \begin{bmatrix} 7 & 4 & | & 22 \ 5 & -9 & | & 15 \end{bmatrix} \].

Key Concepts

System of Linear Equations

Imagine you're trying to solve a puzzle where you're given clues about different numbers. A **system of linear equations** works just like that. It's a collection of two or more equations, all with the same set of variables. In our example:

• The first equation is: \( 7x + 4y = 22 \)
• The second equation is: \( 5x - 9y = 15 \)

Each equation lines up to form straight lines when plotted on a graph. The point where they intersect, if they do, is the solution to the system. This shows you the values that make both equations true simultaneously.

Matrix Representation

When dealing with systems of equations, it's often helpful to use **matrix representation**—a neat, compact way to write everything down. You can think of a matrix as a table of items arranged in rows and columns.For our system:

• The first row comes from the equation \( 7x + 4y = 22 \) and would look like this: \( [7, 4 | 22] \)
• The second row from the equation \( 5x - 9y = 15 \) becomes: \( [5, -9 | 15] \)

The matrix holds all the important information: coefficients of variables and the constants from either side of the equations. It's like keeping all your puzzle clues well-organized before you start solving it!

Linear Algebra Concepts

Linear algebra is all about understanding situations with multiple variables, much like juggling several balls at once. The beauty of **linear algebra concepts** lies in their ability to manage systems of equations efficiently. With these concepts, you can:

• Quickly check if a system of equations has solutions
• Identify if the system has one, many, or no solutions at all
• Simplify a complex system to make solving straightforward

By translating equations into matrices, linear algebra provides powerful methods, like row reduction, to find where solutions lie.

Coefficients Matrix

In any system of equations, coefficients play a central role. The **coefficients matrix** refers to a matrix solely containing the coefficients of variables from the equations. In our example, it would look like:\[\begin{bmatrix}7 & 4 \5 & -9\end{bmatrix}\]The coefficients matrix is essential because:

• It outlines the relationship between variables
• Helps in operations like calculating determinant or inverse, crucial for solutions
• Often simplifies the process of finding solutions to the system

By isolating the coefficients, you can focus purely on how variables interact without any distraction from constants.