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Chapter 4: Problem 33

Solve each system. $$\begin{array}{r}x-y+2 z=3 \\\2 x+y-z=5 \\\3 x-3 y+6 z=4\end{array}$$

Short Answer

Expert verified
The system of equations has no solutions.

Step by step solution

01

Write the system of equations

The system of equations to solve is given as: \[ \begin{array}{rcl} x - y + 2z &=& 3 \ 2x + y - z &=& 5 \ 3x - 3y + 6z &=& 4 \end{array} \]
02

Simplify the system

Notice the third equation is a multiple of the first equation: \[ 3(x - y + 2z) = 3x - 3y + 6z = 3*3 = 9 equation= 3x - 3y + 6z = 9 eq 4 \] Thus, the third equation is contradictory and the system has no solutions.

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Key Concepts

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

linear algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and systems of linear equations. This field is essential for diverse applications in science and engineering. To solve these systems, we often use methods like substitution, elimination, or matrix operations. For example, let's consider our given system: \(\begin{array}{rcl} x - y + 2z &=& 3 \ 2x + y - z &=& 5 \ 3x - 3y + 6z &=& 4 \end{array}\) Understanding how to manipulate and simplify these equations is key to finding solutions.
inconsistent systems
Multiple equations are a central topic in linear algebra. When solving systems of equations, we deal with more than one linear equation simultaneously. The goal is to find values for the variables that make all the equations true. For instance, in our exercise: \(\begin{array}{rcl} x - y + 2z &=& 3 \ 2x + y - z &=& 5 \ 3x - 3y + 6z &=& 4 \end{array}\) Here, we're trying to find values for \(x, y,\) and \(z\) that satisfy all three equations.

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Most popular questions from this chapter

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