Problem 66 - Precalculus

Question

Employ a graphing calculator to solve the system of linear equations (most graphing calculators have the capability of solving linear systems with the user entering the coefficients). $$\begin{array}{rr} 2 x+z+y= & -3 \\ 2 y-z+x= & 0 \\ x+y+2 z= & 5 \end{array}$$

Step-by-Step Solution

Answer

Use a graphing calculator to obtain: \(x = 1\), \(y = -2\), and \(z = 3\).

Step 1: Enter Coefficients into the Calculator

Turn on your graphing calculator and access the function for solving linear systems. You will need to enter the coefficients of each variable from the equations into a matrix format. For the given system, this is a 3x4 augmented matrix:\[\begin{bmatrix}2 & 1 & 1 & | & -3 \1 & 2 & -1 & | & 0 \1 & 1 & 2 & | & 5\end{bmatrix}\] Each row corresponds to one equation, and each column (except the last) corresponds to the coefficients of \(x\), \(y\), and \(z\).

Step 2: Use the Calculator to Solve

Use your calculator's function to perform row operations until the matrix is in reduced row-echelon form, or directly use the calculator's linear system solver function if available. The calculator should perform necessary calculations and yield the solution for the variables.

Step 3: Interpret the Results

Once the calculator has processed the matrix, it will provide either a specific set of values for \(x\), \(y\), and \(z\) or indicate that the system has no unique solution. Based on the calculator's output, you can interpret these results as the solution to the system.

Step 4: Verify the Solution

It's a good practice to check that the calculated values satisfy all the original equations. Substitute \(x\), \(y\), and \(z\) back into the original equations to confirm their validity.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can solve systems of linear equations quickly. These calculators can handle complex calculations and produce results efficiently. To solve a system of equations, like the one given in the exercise, you primarily need to enter the coefficients of the equations.
Most graphing calculators have a mode or function specifically for solving linear systems where you enter these coefficients into a matrix format. After this, the calculator either performs row reduction or uses an internal algorithm to solve the system. The capability to visualize equations graphically is a significant advantage. It provides insight into how the equations interact with each other. If the system has a solution, you may see the point where the lines intersect. If not, the lines may be parallel or identical, indicating infinitely many or no solutions.

Augmented Matrix

An augmented matrix is a crucial concept when dealing with systems of equations. It combines all equations into a single matrix, which consists of the coefficients of each variable in the system and the constants on the right side of the equations.
For the given system:

• Equation 1: Coefficients are 2, 1, and 1, with a constant -3.
• Equation 2: Coefficients are 1, 2, and -1, with constant 0.
• Equation 3: Coefficients are 1, 1, and 2, with constant 5.

These form the augmented matrix:\[\begin{bmatrix}2 & 1 & 1 & | & -3 \1 & 2 & -1 & | & 0 \1 & 1 & 2 & | & 5\end{bmatrix}\]The augmented matrix makes it easier to perform operations and solve for variables systematically. It lays out everything in an organized form, allowing efficient processing either manually or using a calculator.

Row-Reduction

Row-reduction, or Gaussian elimination, is a method to solve systems of linear equations. It involves performing operations on the rows of the augmented matrix to reach the reduced row-echelon form (RREF). In RREF, the matrix has a structure from which you can directly read the solutions.
The process includes:

• Swapping rows if needed to bring non-zero elements to the leading diagonal.
• Multiplying rows by constants to simplify calculations.
• Adding or subtracting rows from each other to eliminate variables step by step.

With a graphing calculator, row-reduction is automated, saving time and reducing errors. The calculator will perform these operations internally and provide you with the final reduced form from which solutions, if they exist, can be easily interpreted.

Solutions Verification

Verifying the solution to a system of linear equations is essential to ensure accuracy, especially when using powerful tools like graphing calculators. Once the calculator provides the solution set, it is crucial to check whether these values actually satisfy the original equations.
To do this, substitute the solution back into each equation:

• Replace the variables in equation 1 with the calculated values and check if the left-hand side equals -3.
• Do the same for equation 2, ensuring it equals 0.
• Finally, verify equation 3's left-hand side results in 5.

If all equations hold true, the solution is correct. This step offers peace of mind and is a good practice whenever you solve mathematical problems, building confidence in the results you obtain.