Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} -60 x+45 y=12 \\ 20 x-15 y=-4 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions, expressed as \(x = \frac{3}{4}t - \frac{1}{5}, y = t\) for any real number \(t\).

Step 1: Write the Augmented Matrix

Convert the given system of equations into an augmented matrix. The system is: \(-60x + 45y = 12\) and \(20x - 15y = -4\). Write it as:\[\begin{bmatrix}-60 & 45 & | & 12 \20 & -15 & | & -4\end{bmatrix}\]

Step 2: Simplify Rows

Simplify the rows to make calculations easier. Divide the first row by 15 and the second row by 5 to get:\[\begin{bmatrix}-4 & 3 & | & rac{4}{5} \4 & -3 & | & -rac{4}{5}\end{bmatrix}\]

Step 3: Eliminate x from Second Equation

Add the first row to the second row to eliminate \(x\) from the second equation:\[\begin{bmatrix}-4 & 3 & | & rac{4}{5} \0 & 0 & | & 0\end{bmatrix}\]

Step 4: Interpret the Result

The resulting matrix corresponds to the equations \(-4x + 3y = \frac{4}{5}\) and \(0 = 0\). The second row, \(0 = 0\), suggests that there are infinitely many solutions as it does not provide any additional information about \(x\) or \(y\).

Step 5: Express Solutions in Parametric Form

Since the second equation gives no new information, express \(x\) in terms of \(y\) using the first equation. Re-arrange \(-4x + 3y = \frac{4}{5}\) to get:\[x = \frac{3}{4}y - \frac{1}{5}\]Expressing in a parametric form, we choose \(y = t\) (a parameter), which means:\(x = \frac{3}{4}t - \frac{1}{5}\), \(y = t\), for all real numbers \(t\).

Key Concepts

Augmented Matrix

The concept of an augmented matrix is a useful tool when solving systems of equations using methods like Gaussian Elimination. An augmented matrix is a compact representation of a system of linear equations, where the coefficients of the variables and the constants from the equations are placed into a matrix format. This format makes it easier to perform row operations and simpler transformations during the elimination process. In our particular exercise, the system of equations is

• \(-60x + 45y = 12\)
• \(20x - 15y = -4\)

The augmented matrix formed from this system is written as:\[\begin{bmatrix}-60 & 45 & | & 12 \20 & -15 & | & -4\end{bmatrix}\]This format compactly includes all the necessary information to perform further operations. The vertical bar indicates the separation between the coefficients of the variables and the constants from the equation. Working with this matrix enables us to apply techniques to systematically reduce the equations until solving for all unknowns.

Parametric Form

When a system of equations possesses infinitely many solutions, expressing the solution in parametric form provides a convenient way to list all possible solutions. In our exercise, the final form of the equations after using Gaussian Elimination resulted in \(-4x + 3y = \frac{4}{5}\)and an equation of \(0 = 0\), indicating infinite solutions.
To express this parametrically, we solve \(-4x + 3y = \frac{4}{5}\)\(x = \frac{3}{4}y - \frac{1}{5}\)By introducing a parameter, say \(t\), to represent one variable, typically the most convenient one, the solutions of the system then become:

• \(x = \frac{3}{4}t - \frac{1}{5}\)
• \(y = t\)

Thus, the parametric form describes all the potential combinations of \(x\) and \(y\) that satisfy the original equations, by varying the parameter \(t\) across all real numbers.

System of Equations

A system of equations is a collection of two or more equations, each containing one or more variables. Solving this system means finding a set of values for the variables that satisfies every equation simultaneously. In our exercise, we were given the system:

• \(-60x + 45y = 12\)
• \(20x - 15y = -4\)

The goal was to use Gaussian Elimination to simplify these equations into a form that makes the solutions more apparent.The Gaussian Elimination method works by transforming the system of equations into an equivalent upper-triangular form, primarily utilizing row operations. This involves using an augmented matrix as an intermediate step to reduce computation complexity and streamline operations. Eventually, through these transformations, you're left with an easy-to-interpret row of equations, revealing the nature of the solutions whether they are unique, infinite, or nonexistent. The understanding of system of equations along with their solutions is foundational in linear algebra and offers practical applications in various fields such as engineering, physics, computer science, and economics.