Question

How do I use Gaussian elimination to solve a system of equations?

Answer 1

In this question, we just have to know how to use Gaussian elimination to solve a system of equations. For solving a system of equations using the Gaussian elimination method, we need to know about how to do elementary operations in a matrix.

Complete step by step answer:
Gauss elimination is the best method to solve the system of equations if somebody does not have a graphing calculator or computer program to help.
The goals of Gaussian elimination are to make the upper left corner element of the matrix as 1, by using elementary operations in a row to get zeroes in all the positions underneath that element 1, get 1 for leading coefficients in every row diagonally from the upper left corner to lower right corner, and get zeroes beneath all leading coefficients. Basically, we eliminate all variables in the last row except for one, all variables except for two in the equation above that one, and so on forth to the top equation, which has all the variables. Then, we can use back substitution to solve for one variable at a time by plugging the values we know into the equations from the bottom up.
We achieve this elimination by eliminating any one variable which comes first in all equations. Then, eliminate the second variable in all equations except for the first two. This process continues, eliminating one more variable per line, until only one variable is left in the last line. Then, solve for that variable.
Some elementary operations are shown here
Let us suppose we have a matrix of $2\times 2$:

4 & 7 \\\ 6 & 5 \\\ \end{matrix} \right)$$ Suppose, $${{r}_{1}}$$ is row 1 and $${{r}_{2}}$$ is row 2. Using elementary operation $$-3{{r}_{2}}\to {{r}_{2}}$$ , we can write $$\left( \begin{matrix} 4 & 7 \\\ 6 & 5 \\\ \end{matrix} \right)\Leftrightarrow \left( \begin{matrix} 4 & 7 \\\ -18 & -15 \\\ \end{matrix} \right)$$ Again, using elementary operation $${{r}_{2}}+{{r}_{1}}\leftrightarrow {{r}_{2}}$$, we can write $$\left( \begin{matrix} 4 & 7 \\\ -18 & -15 \\\ \end{matrix} \right)\Leftrightarrow \left( \begin{matrix} 4 & 7 \\\ -14 & -8 \\\ \end{matrix} \right)$$ Above two were the examples of elementary operations. **Note:** If someone does not have any idea in elementary operation. Then, he/she can get some ideas about that seeing the below operations: We can multiply any by a constant except zero : $$-3{{r}_{2}}\to {{r}_{2}}$$ We can switch any two rows: $${{r}_{2}}\leftrightarrow {{r}_{1}}$$ We can two rows together: $${{r}_{1}}+{{r}_{2}}\to {{r}_{1}}$$