Problem 40 - Contemporary Precalculus

Question

Exercises \(37-40,\) solve the system. [Note: The REF and RREF keys on some calculators produce an error message when there are more rows than columns in a matrix, in which case you will have to solve the system by some other means.] $$\begin{array}{r} x-y=2 \\ x+y=4 \\ 2 x+3 y=9 \\ 3 x-2 y=6 \end{array}$$

Step-by-Step Solution

Answer

Answer: The solution set of the given system of linear equations is {(4, 0)}.

Step 1: Write the system in matrix form

The given system can be written as an augmented matrix: $$\left[ \begin{array}{cc|c} 1 & -1 & 2 \\ 1 & 1 & 4 \\ 2 & 3 & 9 \\ 3 & -2 & 6 \end{array} \right]$$

Step 2: Eliminate x

Add -1 times the first row to the second row, the third row to -2 times the first row, and the fourth row to -3 times the first row to eliminate the x variable, so we get: $$\left[ \begin{array}{cc|c} 1 & -1 & 2 \\ 0 & 2 & 2 \\ 0 & 7 & 5 \\ 0 & 1 & 0 \end{array} \right]$$

Step 3: Solve for y in the last row

The last row indicates that \(y = 0\). Substitute the value of y in the second row to find the value of x: $$2x + 2(0) = 8 \Rightarrow x = 4$$

Step 4: Check if (x, y) works for all the equations

We found the unique solution (x, y) = (4, 0). Now, check if it satisfies all the given equations. Equation 1: \(4-0 = 2\) (True) Equation 2: \(4+0 = 4\) (True) Equation 3: \(2(4)+3(0)=8\) (True) Equation 4: \(3(4)-2(0)=12\) (True) Since the unique solution (x, y) = (4, 0) satisfies all the given equations, the system is consistent and its solution set is: $$\left\{(4,0)\right\}$$

Key Concepts

Matrix Form

When tackling the challenge of solving systems of linear equations, converting the system to a matrix form offers a streamlined pathway to finding solutions. The matrix form reshapes the equation set into a compact, mathematical structure that neatly unpacks each equation into rows.

Imagine each linear equation as a row in a matrix, with the coefficients of the variables occupying the columns. For example, in the equation \(x - y = 2\), the coefficients 1 for \(x\) and -1 for \(y\) are first, and the constant term 2 is last. The augmented matrix further simplifies this form by appending the constants as an additional column, which enables us to perform row operations systematically. The matrix form not only keeps the workspace tidy but also sets the stage for applying powerful techniques such as Gaussian elimination to find solutions.

Row Operations

The manipulation of rows within a matrix, known as row operations, is integral to simplifying a system of equations. These operations include swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting rows. Our goal when applying row operations is to achieve a form where each pivot, or leading entry of a row, reveals a variable's value with minimal effort.

These modifications, though simple, are powerful tools that transform the matrix into an echelon form, a format where each leading coefficient (except in a row of all zeroes) is to the right of the leading coefficient in the row above it. Mastery of row operations ensures that you can strategically and efficiently progress towards a solution without altering the equations' inherent relationships.

Augmented Matrix

An augmented matrix is essentially the matrix form of a system of linear equations with an extension. This extension is an added column that captures the constants from each equation on the right-hand side, reflecting the equations' framework in a single tableau. The 'augmentation' signifies the merger of coefficients and constants into a unified structure.

Known as the last column, this augmentation is separated from the coefficients by a vertical line, making it visually distinct. This form is crucial as it allows for the entire system to be manipulated simultaneously through row operations without losing any information about the equations. The augmented matrix consolidates both the variables and the outcomes they are to achieve, thus encapsulating the complete picture of the system in one place.

Gaussian Elimination

Gaussian elimination is a systematic procedure for resolving systems of linear equations using matrix form and row operations. The technique transforms an augmented matrix into row-echelon form, and ultimately reduced row-echelon form, making it possible to read off solutions directly from the matrix.

This method operates in two main phases: the forward elimination phase reduces the matrix to an upper triangular form, and the backward substitution phase finds the solution by starting from the last row and working upwards. Gaussian elimination is renowned for its efficiency and precision, as it provides clear direction on reducing complex systems into simpler, solvable formats. By systematically applying this process, one can navigate through numerous equations with multiple variables to unveil their solutions with clarity and confidence.