Examples

$x + 5 y = 4 x$ , $x > y$

Step 1

Introduce slack variables $u$ and $v$ to replace the inequalities with equations.

$x - Z = y$

$- 3 x + 5 y = 0$

Step 2

Subtract $y$ from both sides of the equation.

$x - Z - y = 0, - 3 x + 5 y = 0$

Step 3

Write the system of equations in matrix form.

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ - 3 & 5 & 0 & 0 \end{array}]$

Step 4

Find the reduced row echelon form.

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Step 4.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 4.1.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ - 3 + 3 \cdot 1 & 5 + 3 \cdot - 1 & 0 + 3 \cdot 0 & 0 + 3 \cdot 0 \end{array}]$

Step 4.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \end{array}]$

Step 4.2

Multiply each element of $R_{2}$ by $\frac{1}{2}$ to make the entry at $2, 2$ a $1$ .

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Step 4.2.1

Multiply each element of $R_{2}$ by $\frac{1}{2}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ \frac{0}{2} & \frac{2}{2} & \frac{0}{2} & \frac{0}{2} \end{array}]$

Step 4.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}]$

Step 4.3

Perform the row operation $R_{1} = R_{1} + R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 4.3.1

Perform the row operation $R_{1} = R_{1} + R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + 0 \\ 0 & 1 & 0 & 0 \end{array}]$

Step 4.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}]$

Step 5

Use the result matrix to declare the final solutions to the system of equations.

$x = 0$

$y = 0$

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