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Chapter 1: Problem 18

Find the solution set of the system of linear equations represented by the augmented matrix. $$\left[\begin{array}{rrrr} 1 & 2 & 1 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Short Answer

Expert verified
The solution set of the system is {(\(x\), \(y\), \(z\)) = (1 - 2t, t, -1)| t \(\epsilon \) real numbers}

Step by step solution

01

Identify the Equations

The augmented matrix represents the following system of equations:(1) \(1x + 2y + 1z = 0\)(2) \(0x + 0y + 1z = -1\)(3) \(0x + 0y + 0z = 0\)
02

Simplify the Equations

The simplified equations from the augmented matrix are:(1) \(x + 2y + z = 0\), can be written as \(x = -2y - z\)(2) From the Equation (2), \( z = -1\)(3) Equation (3) tells us nothing so can be ignored.
03

Substitute z back into the other equations

Substitute \( z = -1\) into Equation (1)So \(x = -2y - z\)becomes \(x = -2y - (-1)\)which simplifies to \(x = -2y + 1\)
04

Express the Solution Set

Express the solution set for x, y, and z in parametric form using a parameter for y. This gives the full general set of solutions to the system of equations as:\(x = -2y + 1\)\(y = y\)\(z = -1\)We can say \(y = t\) where t is a real number, and write the solution set as: {(\(x\), \(y\), \(z\)) = (1 - 2t, t, -1)| t \(\epsilon \) real numbers}

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Key Concepts

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

Augmented Matrix
In solving systems of linear equations, an augmented matrix is a highly effective tool that represents all the necessary information compactly. It's created by placing the coefficient matrix of the variables next to the constants from the equations' right-hand side. Think of it like a visual snapshot of the system.
For the given exercise, the augmented matrix looks like this:
\[\[\begin{align*} \left[\begin{array}{rrrr}1 & 2 & 1 & 0 \0 & 0 & 1 & -1 \0 & 0 & 0 & 0\end{array}\right]\end{align*}\]\]
Although it might initially look daunting, it's just three rows representing three equations, with the last column reflecting the equal-to part after the

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Most popular questions from this chapter

Determine the size of the matrix. $$\left[\begin{array}{lllll} 1 & 2 & 3 & 0 & 1 \end{array}\right]$$

In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row-echelon form.

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\begin{aligned} x_{1} \quad-3 x_{3}=-2 \\ 3 x_{1}+x_{2}-2 x_{3} =5 \\ 2 x_{1}+2 x_{2}+x_{3} =4 \end{aligned}$$

Does a matrix have a unique row-echelon form? Illustrate your answer with examples. Is the reduced row-echelon form unique?

(a) determine the polynomial function whose graph passes through the given points, and (b) sketch the graph of the polynomial function, showing the given points. $$(-1,3),(0,0),(1,1),(4,58)$$
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