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Chapter 8: Problem 12

Write the augmented matrix for each system of linear equations. $$\begin{aligned} &x-y=-4\\\ &y+z=3 \end{aligned}$$

Short Answer

Expert verified
The augmented matrix is \( \begin{bmatrix} 1 & -1 & 0 & | & -4 \\ 0 & 1 & 1 & | & 3 \end{bmatrix} \).

Step by step solution

01

Write the System of Equations in Standard Form

The system of equations is \( x - y = -4 \) and \( y + z = 3 \). These equations are already in standard form \( ax + by + cz = d \), where terms without variables are isolated on the right side.
02

Identify Coefficients and Constants

For the equation \( x - y = -4 \), the coefficients are 1 (for \(x\)), -1 (for \(y\)), and 0 (for \(z\)) because \(z\) is not explicitly present. The constant \(d\) is -4. For \( y + z = 3 \), the coefficients are 0 (for \(x\)), 1 (for \(y\)), and 1 (for \(z\)), with the constant \(d = 3\).
03

Formulate the Augmented Matrix

The augmented matrix is formed by writing the coefficients of the variables in each equation as a row, followed by the constants in a separate column. Therefore, the augmented matrix is: \[ \begin{bmatrix} 1 & -1 & 0 & | & -4 \ 0 & 1 & 1 & | & 3 \end{bmatrix} \]

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Key Concepts

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

System of Linear Equations
A system of linear equations consists of two or more equations with multiple variables. These equations work together to describe a common solution, meaning a set of variable values that satisfy all equations simultaneously. In our example, we have two equations that need to be solved together:
  • \( x - y = -4 \)
  • \( y + z = 3 \)
These equations can represent many things, from financial calculations to geometric applications. When solving, the aim is to find the values of \( x \), \( y \), and \( z \) that satisfy both equations. This is known as finding the solution to the system of linear equations. Understanding how these equations relate and interact is vital in solving them efficiently.
Standard Form
The standard form for a linear equation is expressed as \( ax + by + cz = d \). This format organizes the equation such that all variables appear on one side, with their respective coefficients, and the constant term on the other side.

In the given exercise:
  • Equation 1: \( x - y = -4 \) is already in standard form, where we can see \( a = 1 \), \( b = -1 \), and \( c = 0 \), with the constant \( d = -4 \).
  • Equation 2: \( y + z = 3 \) follows the standard form as \( a = 0 \), \( b = 1 \), \( c = 1 \), and \( d = 3 \).
Having equations in standard form simplifies the process of identifying coefficients and setting up matrices. It makes comparison and manipulation of equations more straightforward as each component of the equation is explicitly laid out.
Coefficients and Constants
Coefficients are the numerical values that multiply the variables in an equation, essentially telling us how much each variable contributes to the overall equation. In contrast, constants are standalone numbers that do not change, providing the fixed term in the equation.

Let's look at the equations:
  • In \( x - y = -4 \), the coefficients are 1 (for \(x\)), -1 (for \(y\)), and 0 (for \(z\)), while the constant is \(-4\).
  • For \( y + z = 3 \), the coefficients are 0 (for \(x\)), 1 (for \(y\)), and 1 (for \(z\)), with the constant being 3.
Identifying coefficients and constants is crucial as it directly impacts forming the augmented matrix used to solve the system. Recognizing these values allows us to translate the system of equations into a matrix form effectively.
Row Operations
Row operations are mathematical techniques used to simplify matrices in order to solve systems of equations. These operations can include:
  • Swapping two rows.
  • Multiplying a row by a non-zero scalar.
  • Adding or subtracting a multiple of one row to another row.
Such operations are performed to achieve what is called row-echelon or reduced row-echelon form, helping us easily identify the solution to a system of equations.

In this exercise, though we are only forming the augmented matrix, understanding row operations is important for those future steps where one solves the matrix equation. Performing these operations can help simplify complex systems into simpler ones that are more manageable, eventually leading to extracting the values of the variables involved.

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Most popular questions from this chapter

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will eventually reach a maximum height and then fall to the ground. The equation that relates the height \(h\) of a projectile \(t\) seconds after it is projected upward is given by $$h=\frac{1}{2} a t^{2}+v_{0} t+h_{0}$$ where \(a\) is the acceleration due to gravity, \(h_{0}\) is the initial height of the object at time \(t=0,\) and \(v_{0}\) is the initial velocity of the object at time \(t=0 .\) Note that a projectile follows the path of a parabola opening down, so \(a<0\). An object is thrown upward, and the table below depicts the height of the ball \(t\) seconds after the projectile is released. Find the initial height, initial velocity, and acceleration due to gravity. $$\begin{array}{|c|c|} \hline t \text { (seconos) } & \text { Heiant (FEET) } \\ \hline 1 & 54 \\ \hline 2 & 66 \\ \hline 3 & 46 \\ \hline \end{array}$$

Use the following tables. The following table gives fuel and electric requirements per mile associated with gasoline and electric automobiles: $$\begin{array}{|l|c|c|}\hline & \begin{array}{c} \text { Numeer of } \\\\\text { GatLons/Mile }\end{array} & \begin{array}{c}\text { Numeer of } \\\\\text { kW-hr/Mile }\end{array} \\\\\hline \text { SUV full size } & 0.06 & 0 \\ \hline \text { Hybrid car } & 0.02 & 0.1 \\\\\hline \text { Electric car } & 0 & 0.3 \\\\\hline\end{array}$$ The following table gives an average cost for gasoline and electricity: $$\begin{array}{|l|c|}\hline \text { Cost per gallon of gasoline } & \$ 3.80 \\\\\hline \text { Cost per kW-hr of electricity } & \$ 0.05 \\\\\hline\end{array}$$ Environment. Assume you drive 12,000 miles per year. What are the yearly costs associated with driving the three types of cars in Exercise \(81 ?\)

Determine whether each of the following statements is true or false: A system of equations represented by a nonsquare coefficient matrix cannot have a unique solution.

In calculus, when solving systems of linear differential equations with initial conditions, the solution of a system of linear equations is required. solve each system of equations. $$\begin{aligned} &3 c_{1}+3 c_{2}=0\\\ &2 c_{1}+3 c_{2}=0 \end{aligned}$$

$$\begin{aligned} &\text { Verify that } A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\\\-c & a \end{array}\right] \text { is the inverse of }\\\ &A=\left[\begin{array}{ll}a & b \\\c & d\end{array}\right], \text { provided } a d-b c \neq 0 \end{aligned}$$
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