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Section 7-4 : More on the Augmented Matrix

5. For the following system of equations convert the system into an augmented matrix and use the augmented matrix techniques to determine the solution to the system or to determine if the system is inconsistent or dependent.

\[\begin{align*}5x - 25y & = 3\\ - 2x + 10y & = 2\end{align*}\]

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Start Solution

The first step is to write down the augmented matrix for the system of equations.

\[\left[ {\begin{array}{rr|r}5&{ - 25}&3\\{ - 2}&{10}&2\end{array}} \right]\] Show Step 2

We need to make the number in the upper left corner a one. In this case we can do this with the following elementary row operation.

\[\left[ {\begin{array}{rr|r}5&{ - 25}&3\\{ - 2}&{10}&2\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,1}} + \frac{5}{2}{R_{\,2}} \to {R_{\,1}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rr|r}0&0&8\\{ - 2}&{10}&2\end{array}} \right]\] Show Step 3

Okay let’s step back for a second and convert the first row back to an equation. Doing this gives,

\[0 = 8\]

That is clearly not true and we’ve done all our work correctly and so this system is inconsistent and there is no solution to the system.