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Chapter 6: Problem 62

If you could use only one method to solve linear systems in three variables, which method would you select? Explain why this is so.

Short Answer

Expert verified
You might prefer the Gaussian Elimination method for solving linear systems in three variables, because it simplifies the problem through systematic elementary operations without changing the solution, and efficiently handles larger systems and complex calculations.

Step by step solution

01

Determine the Applicable Methods

There are multiple methods applicable for solving linear systems in three variables. Notable among them are Substitution, Elimination, and Gaussian Elimination.
02

Analyze Method

The Substitution method can be time-consuming with larger system of equations and involves more complex calculations. The Elimination method is quicker than substitution, but can also get messy with larger systems.
03

Select and Justify Preferred Method

The Gaussian Elimination method is the preferred method in this case. This is because it is a systematic method that reduces the problem to a simpler form by using elementary operations without changing the solution. Also, it more efficiently handles larger systems and complex calculations.

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Most popular questions from this chapter

Find (if possible) the following matrices: \(a, A B\) \(\boldsymbol{b}, B A\) $$A=\left[\begin{array}{ll}2 & 4 \\\3 & 1 \\\4 & 2\end{array}\right], \quad B=\left[\begin{array}{rrr}3 & 2 & 0 \\\\-1 & -3 & 5\end{array}\right]$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { If } A=\left[\begin{array}{ll}3 & 5 \\\2 & 4\end{array}\right], \text { find}\left(A^{-1}\right)^{-1}$$

Find (if possible) the following matrices: \(a, A B\) \(\boldsymbol{b}, B A\) $$A=\left[\begin{array}{rrrr}2 & -1 & 3 & 2 \\\1 & 0 & -2 & 1\end{array}\right], \quad B=\left[\begin{array}{rr}-1 & 2 \\\1 & 1 \\\3 & -4 \\\6 & 5\end{array}\right]$$

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) is $$\text { Area }-\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right|$$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use determinants to find the area of the triangle whose vertices are \((3,-5),(2,6),\) and \((-3,5)\)

Determinants are used to write an equation of a line passing. through two points. An equation of the line passing through the distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by $$\left|\begin{array}{lll}x & y & 1 \\\x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1\end{array}\right|=0$$ Use the determinant to write an equation of the line passing through \((3,-5)\) and \((-2,6) .\) Then expand the determinant, expressing the line's equation in slope-intercept form.
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