Problem 4 - Contemporary Precalculus

Question

In Exercises \(1-4,\) write the augmented matrix of the system. $$\begin{aligned} x+7 y-\frac{2}{5} z+\frac{5}{6} w &=0 \\ \frac{1}{8} x-y-8 z &=1 \\ \frac{2}{3} y-5 z+\quad w &=-2 \\ \frac{1}{6} x+4 y+\frac{2}{7} z &=3 \end{aligned}$$

Step-by-Step Solution

Answer

Question: Write the augmented matrix of the following system of linear equations: \(x + 7y -\frac{2}{5}z + \frac{5}{6}w = 0\) \(\frac{1}{8}x - y - 8z = 1\) \(\frac{2}{3}y - 5z + w = -2\) \(\frac{1}{6}x + 4y + \frac{2}{7}z = 3\) Answer: The augmented matrix of the given system is: $$ \begin{bmatrix} 1 & 7 & -\frac{2}{5} & \frac{5}{6} & 0 \\ \frac{1}{8} & -1 & -8 & 0 & 1 \\ 0 & \frac{2}{3} & -5 & 1 & -2 \\ \frac{1}{6} & 4 & \frac{2}{7} & 0 & 3 \end{bmatrix} $$

Step 1: Identify coefficients for each variable in each equation

Check every variable for each equation, and write down the corresponding coefficient: For the first equation: \(x = 1, y = 7, z = -\frac{2}{5}, w = \frac{5}{6}\) For the second equation: \(x = \frac{1}{8}, y = -1, z = -8, w = 0\) For the third equation: \(x = 0, y = \frac{2}{3}, z = -5, w =1\) For the fourth equation: \(x = \frac{1}{6}, y = 4, z = \frac{2}{7}, w = 0\)

Step 2: Put the coefficients into the augmented matrix

The matrix will have 4 rows and 5 columns. The first four columns represent the coefficients of variables x, y, z, and w, respectively. The last column represents constants on the right side of each equation. The augmented matrix will look like this: $$ \begin{bmatrix} 1 & 7 & -\frac{2}{5} & \frac{5}{6} & 0 \\ \frac{1}{8} & -1 & -8 & 0 & 1 \\ 0 & \frac{2}{3} & -5 & 1 & -2 \\ \frac{1}{6} & 4 & \frac{2}{7} & 0 & 3 \end{bmatrix} $$

Key Concepts

Linear Equations

A linear equation is an important type of algebraic expression. It is called "linear" because it can be represented graphically as a straight line. Linear equations involve variables raised only to the first power. In these equations, each term is either a constant or the product of a constant and a single variable. Consider an equation like this one: \(2x + 3y = 6\). Here, \(x\) and \(y\) are variables, and their coefficients are 2 and 3, respectively.

• Variables: Values that can change, represented by letters such as \(x\), \(y\), or \(z\).
• Coefficients: Fixed numbers that multiply the variables.
• Constant: A number without a variable part, like the 6 in the equation above.

Knowing how to manipulate and solve linear equations is foundational for algebra. It helps in understanding more complex equations and different systems of equations.

System of Equations

A system of equations consists of multiple equations that share variables. Solving a system means finding values that satisfy all equations simultaneously. Imagine you need to find a point of intersection for two lines on a graph. That point is your solution to the system of equations consisting of those two linear equations.Systems can be:

• Consistent: Having at least one solution.
• Inconsistent: Having no solutions.
• Dependent: Infinite solutions because they represent the same line.

For example, in a system:\[\begin{align*}2x + y &= 5 \3x - y &= 4\end{align*}\]You may use methods like substitution, elimination, or matrix representation to solve the system and find the values of \(x\) and \(y\) that make both equations true.

Matrix Representation

Matrix representation is a way of expressing a system of linear equations in a compact form. Instead of writing each equation separately, you can condense the equations into a matrix. This method makes solving complex systems more manageable.An augmented matrix is created by arranging the coefficients of the variables and the constants from each equation in rows and columns. In our original exercise, we arranged the coefficients and constants from the linear equations into a matrix:\[\begin{bmatrix}1 & 7 & -\frac{2}{5} & \frac{5}{6} & 0 \\frac{1}{8} & -1 & -8 & 0 & 1 \0 & \frac{2}{3} & -5 & 1 & -2 \\frac{1}{6} & 4 & \frac{2}{7} & 0 & 3\end{bmatrix}\]

• The first four columns represent the coefficients of the variables \(x\), \(y\), \(z\), and \(w\).
• The last column contains the constants from the right side of the equations.

Using matrices simplifies the solution process and helps when using computational tools for more complex scenarios.