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Chapter 9: Problem 36

Solve the system of equations using Gaussian elimination or Gauss-Jordan elimination. Use a graphing calculator to check your answer. $$\begin{aligned} m+n+t &=9 \\ m-n-t &=-15 \\ 3 m+n+t &=2 \end{aligned}$$

Short Answer

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Step by step solution

01

- Set up the augmented matrix

Write the given system of equations as an augmented matrix: \[ \begin{pmatrix} 1 & 1 & 1 & | & 9 \ 1 & -1 & -1 & | & -15 \ 3 & 1 & 1 & | & 2 \end{pmatrix} \]
02

- Perform row operations to obtain zeros below the first entry of the first column

Subtract the first row from the second row and subtract 3 times the first row from the third row to get: \[ \begin{pmatrix} 1 & 1 & 1 & | & 9 \ 0 & -2 & -2 & | & -24 \ 0 & -2 & -2 & | & -25 \end{pmatrix} \]
03

- Simplify the second row

Divide the second row by -2 to simplify: \[ \begin{pmatrix} 1 & 1 & 1 & | & 9 \ 0 & 1 & 1 & | & 12 \ 0 & -2 & -2 & | & -25 \end{pmatrix} \]
04

- Eliminate the y-values below the pivot in the second column

Add 2 times the second row to the third row: \[ \begin{pmatrix} 1 & 1 & 1 & | & 9 \ 0 & 1 & 1 & | & 12 \ 0 & 0 & 0 & | & -1 \end{pmatrix} \]
05

- Interpret the results

Since the third row is \[ 0 = -1 \], we identify that the system is inconsistent and there is no solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

systems of equations
A system of equations is a collection of two or more equations with the same set of unknowns. In this exercise, we have three equations involving three unknowns: m, n, and t. The goal is to find values for these variables that satisfy all equations simultaneously. Systems of equations can be solved using various methods such as substitution, elimination, or matrix approaches like Gaussian elimination.
augmented matrix
An augmented matrix is a compact way of representing a system of linear equations. Instead of writing each equation separately, we write the coefficients and the constants in a matrix format. For example, the system of equations in our exercise is represented as:
\[ \begin{pmatrix} 1 & 1 & 1 & | & 9 \ 1 & -1 & -1 & | & -15 \ 3 & 1 & 1 & | & 2 \ \ end{pmatrix} \] Here, the vertical line separates the coefficients from the constants on the right-hand side of each equation. This matrix helps in performing row operations systematically to solve the system.
inconsistent system
An inconsistent system is one that has no solutions. This happens when the equations contradict each other. In our exercise, we reach the last row of the matrix which indicates \[ 0 = -1 \]. Since this is a clear contradiction, it tells us that the system has no solution. Therefore, no set of values for m, n, and t will simultaneously satisfy all three equations.

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Most popular questions from this chapter

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