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Chapter 9: Problem 2

Write each system in an augmented matrix. $$\begin{array}{c}x+6 y=4 \\\\-5 x+y=-3\end{array}$$

Short Answer

Expert verified
The short answer for the given system of linear equations is: \[ \left[ \begin{array}{cc|c} 1 & 6 & 4 \\ -5 & 1 & -3 \end{array} \right] \]

Step by step solution

01

Identify the coefficients and constants

First, let's identify the coefficients of the variables and the constants in the given linear equations: Equation 1: \(x + 6y = 4\) Coefficients: \(1\) for \(x\) and \(6\) for \(y\), Constant: \(4\) Equation 2: \(-5x + y = -3\) Coefficients: \(-5\) for \(x\) and \(1\) for \(y\), Constant: \(-3\)
02

Arrange the coefficients and constants in a matrix

Now, we arrange the coefficients and constants in a matrix form. The first column represents the coefficients of the variable \(x\), the second column represents the coefficients of the variable \(y\), and the third column represents the constant terms. Thus, the augmented matrix for the given system of linear equations will be: $$ \left[ \begin{array}{cc|c} 1 & 6 & 4 \\ -5 & 1 & -3 \end{array} \right] $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are mathematical expressions involving one or more variables where the variables are only raised to the power of one. In simpler terms, they are equations that form straight lines when graphed on a coordinate plane. A typical form of a linear equation with two variables, such as in our example, is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

In the given problem, our system consists of two linear equations:
  • \(x + 6y = 4\)
  • \(-5x + y = -3\)
Each equation in this system represents a line on a graph, and the solution to these equations is the point where the lines intersect.
Matrix Representation
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of linear equations, a matrix can succinctly represent the system of equations, making it easier to perform calculations. The matrix form can help us use methods like row reduction to find solutions.

For linear equations, we often use an augmented matrix. The augmented matrix includes the coefficients of the variables and the constants from each equation. This form is shown with a vertical line dividing the coefficients and the constants, making it easier to separate the elements that make up the total equation system.
Coefficients
Coefficients are the numbers that multiply the variables in an equation. They indicate how much the variable (like \(x\) or \(y\)) contributes to the entire equation. Identifying coefficients is important because they form the main structure of an augmented matrix.

In our given equations:
  • For \(x + 6y = 4\), the coefficients are \(1\) and \(6\).
  • For \(-5x + y = -3\), the coefficients are \(-5\) and \(1\).
These numbers are placed in the matrix, directly helping us understand and manipulate the system to find solutions.
Constants
Constants in an equation are the standalone numbers that do not accompany a variable. They represent fixed values which, in the context of linear equations, often denote the point where a line crosses the y-axis when the equation is graphed.

In the process of creating an augmented matrix, constants are crucial as they fill the matrix's last column. From our equations:
  • in \(x + 6y = 4\), the constant is \(4\).
  • in \(-5x + y = -3\), the constant is \(-3\).
These numbers are vital in solving the equations as they affect the entire outcome of the system.

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Most popular questions from this chapter

Write a system of linear equations in \(x, y,\) and \(z\) represented by each augmented matrix. $$\left[\begin{array}{rrr|r}1 & 4 & -3 & -5 \\\\-1 & 2 & 5 & 8 \\\6 & -2 & -1 & 3\end{array}\right]$$

Graph each compound inequality. \(2 x+5 y<15\) or \(y \leq \frac{3}{4} x-1\)

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$|6 k+17|>-4$$

Graph using either the test point or slope-intercept method. \(6 x-y \leq 2\)

Write a system of linear equations in \(x, y,\) and \(z\) represented by each augmented matrix. $$\left[\begin{array}{rrr|r}1 & -3 & 2 & 7 \\\4 & -1 & 3 & 0 \\\\-2 & 2 & -3 & -9\end{array}\right]$$
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