Algebra Examples

$3 x + 1 = 0 y$ , $y + 2 x > - 9$

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

$y + 2 x - Z = - 9$

$3 x + 1 = 0$

Move $1$ to the right side of the equation because it does not contain a variable.

$y + 2 x - Z = - 9$

$3 x = - 1$

Write the system of equations in matrix form.

$[\begin{array}{c} 2 & 1 & 0 & - 9 \\ 3 & 0 & 0 & - 1 \end{array}]$

Row reduce.

Tap for more steps...

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

Tap for more steps...

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

$[\begin{array}{c} \frac{1}{2} R_{1} & \frac{1}{2} R_{1} & \frac{1}{2} R_{1} & \frac{1}{2} R_{1} \\ 3 & 0 & 0 & - 1 \end{array}]$

$R_{1} = \frac{1}{2} R_{1}$

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

$[\begin{array}{c} (\frac{1}{2}) \cdot (2) & (\frac{1}{2}) \cdot (1) & (\frac{1}{2}) \cdot (0) & (\frac{1}{2}) \cdot (- 9) \\ 3 & 0 & 0 & - 1 \end{array}]$

$R_{1} = \frac{1}{2} R_{1}$

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 3 & 0 & 0 & - 1 \end{array}]$

Perform the row operation $R_{2} = - 3 \cdot R_{1} + R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $0$ .

Tap for more steps...

Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - 3 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} \end{array}]$

$R_{2} = - 3 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 3 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ (- 3) \cdot (1) + 3 & (- 3) \cdot (\frac{1}{2}) + 0 & (- 3) \cdot (0) + 0 & (- 3) \cdot (- \frac{9}{2}) - 1 \end{array}]$

$R_{2} = - 3 \cdot R_{1} + R_{2}$

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & - \frac{3}{2} & 0 & \frac{25}{2} \end{array}]$

Perform the row operation $R_{2} = - \frac{2}{3} R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $1$ .

Tap for more steps...

Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - \frac{2}{3} R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ - \frac{2}{3} R_{2} & - \frac{2}{3} R_{2} & - \frac{2}{3} R_{2} & - \frac{2}{3} R_{2} \end{array}]$

$R_{2} = - \frac{2}{3} R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - \frac{2}{3} R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ (- \frac{2}{3}) \cdot (0) & (- \frac{2}{3}) \cdot (- \frac{3}{2}) & (- \frac{2}{3}) \cdot (0) & (- \frac{2}{3}) \cdot (\frac{25}{2}) \end{array}]$

$R_{2} = - \frac{2}{3} R_{2}$

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

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

Tap for more steps...

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

$[\begin{array}{c} - \frac{1}{2} R_{2} + R_{1} & - \frac{1}{2} R_{2} + R_{1} & - \frac{1}{2} R_{2} + R_{1} & - \frac{1}{2} R_{2} + R_{1} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

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

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

$[\begin{array}{c} (- \frac{1}{2}) \cdot (0) + 1 & (- \frac{1}{2}) \cdot (1) + \frac{1}{2} & (- \frac{1}{2}) \cdot (0) + 0 & (- \frac{1}{2}) \cdot (- \frac{25}{3}) - \frac{9}{2} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

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

$x = 0$

$y = 0$

Enter YOUR Problem