Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, write the augmented matrix for the linear system. $$ \begin{array}{c} 16 y=4 \\ 9 x-y=2 \end{array} $$

Step-by-Step Solution

Answer

The augmented matrix is \(\begin{bmatrix} 0 & 16 & | & 4 \\ 9 & -1 & | & 2 \end{bmatrix}\).

Step 1: Write the system of equations

The given linear system is:1. \(16y = 4\)2. \(9x - y = 2\)

Step 2: Arrange equations into standard form

Rewrite the equations into the standard form \(Ax + By = C\):1. For \(16y = 4\), since there is no \(x\) term, it becomes \(0x + 16y = 4\).2. The equation \(9x - y = 2\) is already in standard form.

Step 3: Identify coefficients and constant terms

From the standard form equations:1. The coefficients for the first equation \(0x + 16y = 4\) are \(A=0\), \(B=16\), and \(C=4\).2. The coefficients for the second equation \(9x - y = 2\) are \(A=9\), \(B=-1\), and \(C=2\).

Step 4: Write the augmented matrix

Construct the augmented matrix using the coefficients and constant terms from each equation:\[\begin{bmatrix}0 & 16 & | & 4 \9 & -1 & | & 2\end{bmatrix}\]

Key Concepts

Linear System

When we talk about a **linear system**, we're referring to a collection of linear equations that involve the same set of variables. Linear equations can have variables like \(x\), \(y\), or more, and they all join to form a system. The goal is to find the values of the variables that satisfy all the equations at the same time. Think of it like trying to solve a puzzle where you need to find the number for each letter that makes every piece fit.

Example: In a puzzle involving \(16y = 4\) and \(9x - y = 2\), you are trying to find values of \(x\) and \(y\) that make both equations true simultaneously. Whether there are two equations, three, or more, as long as they're capable of being written in a linear format, it's a linear system.

Why is this important? Solving a linear system allows you to understand relationships between variables and is fundamental in many mathematical modeling tasks.

System of Equations

A **system of equations** is essentially what it sounds like: a few equations that need to work together in harmony. These are usually written together to indicate they share common solutions.

Here's how it works with the example given:

• Equation 1: \(16y = 4\)
• Equation 2: \(9x - y = 2\)

Both of these equations are linked because we're trying to find the same values for \(x\) and \(y\) that satisfy both.

You start solving these by simplifying or rearranging them, aiming to find the missing values. In practice, you could add them together, subtract them, or use substitutions so they can reveal the hidden solution.

Matrix Representation

**Matrix representation** of a linear system of equations is a neat way to organize all your numbers into a structured layout. It helps you to efficiently carry out complex calculations. When representing the equations \(16y = 4\) and \(9x - y = 2\), we first rewrite them ensuring every term is clear, recognizing any missing pieces.

For example, in the equation \(16y = 4\), there's no \(x\) term, so we represent it as \(0x + 16y = 4\). This gives it consistency when lining it up against the second equation.

To form the **augmented matrix**, the coefficients and constants are used to create a rectangular array:

• First equation coefficients: \( [0, 16, |, 4] \)
• Second equation coefficients: \( [9, -1, |, 2] \)

This results in the matrix type form, visually represented as:\[\begin{bmatrix}0 & 16 & | & 4 \9 & -1 & | & 2\end{bmatrix}\]This matrix opens the door to using matrix operations which can simplify solving the whole system, especially as the number of equations grows.