X = 0.087

Y = 2.6087

Z = -1.1304

CALCULATE

CALCULATE

** Gauss Elimination Calculator** solve a system of three linear equations with real coefficients using Gaussian elimination algorithm. It is an online algebra tool programmed to determine an ordered triple as a solution to a system of three linear equations. Using this calculator, we will able to understand how to solve the system of linear equations using Gauss elimination algorithm.

It is necessary to follow the next steps:

- Enter twelve coefficients of a system of linear equations in the box. These coefficients must be real numbers.
- Press the "Generate Work" button to make the computation;
__Gauss Elimination Calculator__will give an ordered triple $(x,y,z)$ as a solution of a system of three linear equations.

__Gauss elimination__ or __row reduction__, is an algorithm for solving a system of linear equations. This method also called as Gauss-Jordan elimination. It is represented by a sequence of operations performed on the matrix. The method is named after Carl Friedrich Gauss (1777-1855), although it was known to Chinese mathematicians.
The method of solving a system of linear equations by Gauss elimination is similar to the method of solving matrices. For instance, there is the connection between a system of three linear equations and its coefficient matrix.
$$\begin{align} &a_1x+b_1y+c_1z={ d_1}\\
&a_2x+b_2y+c_2z={ d_2}\\
&a_3x+b_3y+c_3z={ d_3}\\
\end{align} \quad\longmapsto \left(
\begin{array}{ccc}
{a_1} & b_1 &c_1\\
{a_2} &b_2 &c_2\\
{a_3} &b_3 &c_3\\
\end{array}
\right)$$
There are three types of elementary row operations:

- Replacing two rows;
- Multiplying a row by a nonzero number;
- Adding a multiple of one row to another row.

- The leading coefficient of each row must be $1$;
- All elements in a column below a leading $1$ must be $0$;
- All rows that contain zeros are at the bottom of the matrix.

- Divide row $1$ by $4$ ($R_1=\frac {R_1}4)$, to get $$\left( \begin{array}{ccc|c} 1 & \frac 54 &\frac 34&\frac{5}2\\ 3 &6 &7&8\\ 2 &3&0&8\\ \end{array} \right)$$
- Subtract row $1$ multiplied by $3$ from row $2$ ($R_2=R_2-3R_1$), to get $$\left( \begin{array}{ccc|c} 1 & \frac 54 &\frac 34&\frac{5}2\\ 0 &\frac 94 &\frac{19}4&\frac 12\\ 2 &3&0&8\\ \end{array} \right)$$
- Subtract row $1$ multiplied by $2$ from row $3$ ($R_3=R_3-2R_1$), to get $$\left( \begin{array}{ccc|c} 1 & \frac 54 &\frac 34&\frac{5}2\\ 0 &\frac 94 &\frac{19}4&\frac 12\\ 0 &\frac12 &-\frac 32&3\\ \end{array} \right)$$
- Multiply row $2$ by $\frac 49$ ($R_2=\frac49 R_2$), to get $$\left( \begin{array}{ccc|c} 1 & \frac 54 &\frac 34&\frac{5}2\\ 0 &1 &\frac{19}9&\frac 29\\\ 0 &\frac12 &-\frac 32&3\\ \end{array} \right)$$
- Subtract row $2$ multiplied by $\frac 54$ from row $1$ ($R_1=R_1-\frac54 R_2$), to get $$\left( \begin{array}{ccc|c} 1 & 0 &-\frac {17}9&\frac{20}9\\ 0 &1 &\frac{19}9&\frac 29\\ 0 &\frac12 &-\frac 32&3\\ \end{array} \right)$$
- Subtract row $2$ multiplied by $\frac 12$ from row $3$ ($R_3=R_3-\frac12R_2$), to get $$\left( \begin{array}{ccc|c} 1 & 0 &-\frac {17}9&\frac{20}9\\ 0 &1 &\frac{19}9&\frac 29\\ 0 &0&-\frac{23}9&\frac{26}9\\ \end{array} \right)$$
- Multiply row $3$ by $-\frac9{23}$ ($R_3=-\frac9{23}R_3$), to get $$\left( \begin{array}{ccc|c} 1 & 0 &-\frac {17}9&\frac{20}9\\ 0 &1 &\frac{19}9&\frac 29\\ 0 &0&1&-\frac{26}{23}\\ \end{array} \right)$$
- Add row $3$ multiplied by $\frac{17}9$ to row $1$ ($R_1=R_1+\frac{17}9R_3$), to get $$\left( \begin{array}{ccc|c} 1 & 0 &0&\frac2{23}\\ 0 &1 &\frac{19}9&\frac 29\\ 0 &0&1&-\frac{26}{23}\\ \end{array} \right)$$
- Subtract row $3$ multiplied by $\frac {19}9$ from row $2$ ($R_2=R_2-\frac{19}9R_3$), to obtain $$\left( \begin{array}{ccc|c} 1 & 0 &0&\frac2{23}\\ 0 &1 &0&\frac {60}{23}\\ 0 &0&1&-\frac{26}{23}\\ \end{array} \right)$$ So the solution of the system is $(x, y, z) = (\frac{2}{23},\frac{60}{23}, -\frac{26}{23})$.

Gaussian elimination algorithm is useful for determining the rank of a matrix (an important property of each matrix). This method can also help us to find the inverse of a matrix. In Geometry, the equation $Ax+By+Cz=D$ defines a plane in the three-dimensional coordinate system. If we consider a system of three variables, we can think about the points of intersection of planes. Hence, we can determine whether planes are parallel, intersect each other or coincide.

**Practice Problem 1:**

Using the Gauss elimination, solve the system of equations
$$\begin{align} &2x+4y-z=-1\\
&x+3y+7z=2\\
&x+2y+z=-5\\
\end{align} $$
**Practice Problem 2:**

A math library wants to purchase $25$ books for $\$2,800$. Three different types of books are available: a geometry with a price of $\$35$, an algebra with a price of $\$70$, a statistics with a price of $\$140$. How many of each type of books should the library purchase?

The Gauss Elimination Calculator, formula, example calculation (work with steps) and practice problems would be very useful
for grade school students of K-12 education to understand the concept of solving systems of linear equations. This concept is conceived in almost all areas of science, so it will be helpful in solving more complex problems.