Algebra Examples

$x - 2 y = 93$ , $x - y > 0$

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

$x - y - Z = 0$

$x - 2 y - 93 = 0$

Add $93$ to both sides of the equation.

$x - 2 y = 93$

$x - y - Z = 0$

Write the system of equations in matrix form.

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ 1 & - 2 & 0 & 93 \end{array}]$

Find the reduced row echelon form of the matrix.

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Perform the row operation $R_{2} = - 1 \cdot R_{1} + R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - 1 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

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

$R_{2} = - 1 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 1 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ (- 1) \cdot (1) + 1 & (- 1) \cdot (- 1) - 2 & (- 1) \cdot (0) + 0 & (- 1) \cdot (0) + 93 \end{array}]$

$R_{2} = - 1 \cdot R_{1} + R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ 0 & - 1 & 0 & 93 \end{array}]$

Perform the row operation $R_{2} = - 1 \cdot R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $1$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - 1 \cdot R_{2}$ in order to convert some elements in the row to the desired value $1$ .

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

$R_{2} = - 1 \cdot R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 1 \cdot R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ (- 1) \cdot (0) & (- 1) \cdot (- 1) & (- 1) \cdot (0) & (- 1) \cdot (93) \end{array}]$

$R_{2} = - 1 \cdot R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ 0 & 1 & 0 & - 93 \end{array}]$

Perform the row operation $R_{1} = R_{2} + R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} R_{2} + R_{1} & R_{2} + R_{1} & R_{2} + R_{1} & R_{2} + R_{1} \\ 0 & 1 & 0 & - 93 \end{array}]$

$R_{1} = R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = R_{2} + R_{1}$ .

$[\begin{array}{c} 0 + 1 & 1 - 1 & 0 + 0 & - 93 + 0 \\ 0 & 1 & 0 & - 93 \end{array}]$

$R_{1} = R_{2} + R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & 0 & 0 & - 93 \\ 0 & 1 & 0 & - 93 \end{array}]$

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

$x = 0$

$y = 0$