Problem 34

Question

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{rrr|r} 1 & 0 & 1 & 50 \\ 1 & 1 & 0 & 20 \\ 0 & 1 & 1 & -90 \end{array}\right] $$

Step-by-Step Solution

Verified

Answer

The solution is $ x = 80 $, $ y = -60 $, and $ z = -30 $.

1Step 1: Write the System of Equations

First, translate the augmented matrix into a system of linear equations. The given matrix corresponds to these equations: $ x + 0y + z = 50 $ $ x + y + 0z = 20 $ $ 0x + y + z = -90 $.

2Step 2: Use Row Operations to Simplify

To eliminate variables using Gaussian elimination, we first subtract the first equation from the second to make the coefficient of $x$ in the second equation zero: $ (x + y) - (x + z) = 20 - 50 $ simplifies to: $ y - z = -30 $. So, our new system is: $ x + z = 50 $; $ y - z = -30 $; $ y + z = -90 $.

3Step 3: Solve for One Variable

Let's solve the second equation for $y$: $ y = z - 30 $. Now substitute $y = z - 30 $ into the third equation: $ (z - 30) + z = -90 $. Solving this gives $ 2z - 30 = -90 $.

4Step 4: Find Z

Solve the equation $ 2z - 30 = -90 $: Add 30 to both sides: $ 2z = -60 $. Divide both sides by 2: $ z = -30 $.

5Step 5: Find Y

Substitute $ z = -30 $ back into $ y = z - 30 $: $ y = -30 - 30 = -60 $.

6Step 6: Find X

Substitute $ z = -30 $ into the first equation $ x + z = 50 $: $ x - 30 = 50 $. Add 30 to both sides to solve for $ x $: $ x = 80 $.

7Step 7: Verify Solution

Substitute $ x = 80 $, $ y = -60 $, $ z = -30 $ into the original set of equations to verify that all equations hold true. All calculations hold, confirming that these values satisfy the system.

Key Concepts

System of Linear EquationsAugmented MatrixRow OperationsSolution Verification

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. In most cases, people are trying to find the values of these variables that make all the equations true simultaneously.
This exercise involves three equations with three variables: $ x $, $ y $, and $ z $. Each equation represents a line or plane, and solving for the variables finds the intersection point of these geometrical shapes. Here, the equations were extracted from the augmented matrix and rephrased as:

$ x + z = 50 $
$ x + y = 20 $
$ y + z = -90 $

Understanding the interaction among these equations is key to solving them.

Augmented Matrix

An augmented matrix is a handy tool for solving systems of linear equations. It includes both the coefficients of the variables and the constants from the equations on the right.
In this exercise, the augmented matrix is:\[\begin{bmatrix} 1 & 0 & 1 & | & 50 \1 & 1 & 0 & | & 20 \0 & 1 & 1 & | & -90 \end{bmatrix}\]
This matrix format simplifies the visualization of the coefficients and constants, allowing for easier manipulation and solution finding using row operations.
The vertical line often indicates the equality sign in the original equations. Use the matrix to perform operations that help bring it to a form where the solution can be read directly.

Row Operations

Row operations are essential steps in Gaussian elimination or any matrix solution method. These operations help in transforming the matrix into simpler forms to extract the solution.
There are three primary row operations for this purpose:

Swap: Exchange two rows if required.
Scale: Multiply a row by a non-zero constant.
Replace: Add or subtract multiples of rows from each other.

In this exercise, by applying these operations systematically, we simplify the system step by step.
For example, subtracting one row from another to eliminate variables is a crucial move in simplifying the equations. These operations transform the original matrix into a simpler matrix, from which the solution can be more easily deduced.

Solution Verification

Verifying a solution is an integral step to ensure the accuracy of your solution. Once you've found values for the variables, substitute them back into the original equations.
In our case, solving gave us