Answered: Use the Gauss-Jordan method to solve…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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Use the Gauss-Jordan method to solve the following system of equations.

4x-3y=7

8x-6y=14

Select the correct choice below and answer if necessary.

A. Ther solution is ___. (type a simplified ordered pair)

b. There are infinitely many solutions. The solution is (__,y) where y is any real number. (Simplify your answer. Use integers or fractions for any numbers in the expression.)

C. There is no solution.

Expert Solution

Step 1

write the system of equations in the matrix form

$[\begin{matrix} 4 & - 3 \\ 8 & - 6 \end{matrix} \begin{matrix} 7 \\ 14 \end{matrix}]$

Operate $R_{1} \to \frac{1}{4} R_{1}$

$[\begin{matrix} 1 & - \frac{3}{4} \\ 8 & - 6 \end{matrix} \begin{matrix} \frac{7}{4} \\ 14 \end{matrix}]$

operate $R_{2} \to R_{2} - 8 R_{1}$

$[\begin{matrix} 1 & - \frac{3}{4} \\ 0 & 0 \end{matrix} \begin{matrix} \frac{7}{4} \\ 0 \end{matrix}]$

The solution is of the form $A X = 0$

$[\begin{matrix} 1 & - \frac{3}{4} \\ 0 & 0 \end{matrix} \begin{matrix} \frac{7}{4} \\ 0 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]$