Problem 12 - A Graphical Approach to College Algebra

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} &4 x-2 y+3 z=4\\\ &3 x+5 y+z=7\\\ &5 x-y+4 z=7 \end{aligned}$$

Step-by-Step Solution

Answer

Augmented matrix: \[\begin{bmatrix}4 & -2 & 3 & | & 4 \\ 3 & 5 & 1 & | & 7 \\ 5 & -1 & 4 & | & 7 \end{bmatrix}\].

Step 1: Write the Augmented Matrix

For the given system of linear equations, each equation can be represented by the coefficients of the variables and the constant term on the right side. Write these coefficients and constants in matrix form, with each row representing one equation. The matrix for our system becomes: \[\begin{bmatrix}4 & -2 & 3 & | & 4 \3 & 5 & 1 & | & 7 \5 & -1 & 4 & | & 7 \\end{bmatrix}\]

Step 2: Set up the problem

Write out the given matrices or vectors in standard form.

Step 3: Perform the matrix operations

Execute the required operations, showing key intermediate steps.

Step 4: Analyze the result

Interpret the result in terms of the original problem.

Step 5: State the conclusion

Clearly state the final answer.

Step 6: Conclude with the answer

Augmented matrix: \[\begin{bmatrix}4 & -2 & 3 & | & 4 \\ 3 & 5 & 1 & | & 7 \\ 5 & -1 & 4 & | & 7 \end{bmatrix}\].

Key Concepts

Linear Equations

A linear equation is an equation where each term is either a constant or the product of a constant and a single variable. They form the foundation of various mathematical studies including algebra and calculus. Linear equations describe lines in two-dimensional space and planes in three-dimensional space. This makes them quite versatile for modeling real-world problems.

In a linear equation, such as the ones you are working with, the highest power of the variable is always 1. This linearity results in straightforward relationships between variables, unlike quadratic or polynomial equations which have more complex characteristics.

• The standard form of a linear equation in three variables is: \(ax + by + cz = d\).
• Each variable is raised only to the first power, making the solution process simpler.

Understanding linear equations is crucial as they often serve as building blocks for more advanced mathematical concepts.

Coefficients

In the context of linear equations, coefficients are the numerical factors that multiply the variables. They play a crucial role in defining the equation. For example, in the equation \(4x - 2y + 3z = 4\), the numbers 4, -2, and 3 are the coefficients of variables \(x\), \(y\), and \(z\), respectively.

Coefficients determine the slope or behavior of the line or plane described by the equation. They can also provide information on how changes in one variable affect others.

• Positive coefficients indicate a direct relationship, where an increase in the variable leads to an increase in the equation’s outcome.
• Negative coefficients suggest an inverse relationship, where an increase in the variable results in a decrease in the outcome.

Grasping the idea of coefficients helps you understand and manipulate linear equations to solve them accurately.

Matrix Representation

A matrix is an organized rectangular array of numbers, symbols, or expressions. It represents a system of equations in a compact, efficient form. This is particularly useful for solving systems of equations using methods like Gaussian elimination or matrix algebra.

For the system of equations you are dealing with, a matrix representation simplifies handling the data. Matrix representation involves listing the coefficients of the variables in each equation along with the constants from the right side.

In augmented matrices, a vertical bar separates the coefficients from the constant terms. This helps in distinguishing the equations from each solution they might yield:

• The matrix you formed is known as an ">augmented matrix," essential in representing linear systems.
• Each row corresponds to a single equation, and each column corresponds to a particular variable or constant.

Understanding matrices paves the way for advanced problem-solving techniques, making them a quintessential part of mathematics learning.