Algebra Examples

$2 x + 2 y = 6$ , $x + 2 y > 9$

Step 1

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

$x + 2 y - Z = 9$

$2 x + 2 y - 6 = 0$

Step 2

Add $6$ to both sides of the equation.

$x + 2 y - Z = 9, 2 x + 2 y = 6$

Step 3

Write the system of equations in matrix form.

$[\begin{array}{c} 1 & 2 & 0 & 9 \\ 2 & 2 & 0 & 6 \end{array}]$

Step 4

Find the reduced row echelon form.

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Step 4.1

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

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Step 4.1.1

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

$[\begin{array}{c} 1 & 2 & 0 & 9 \\ 2 - 2 \cdot 1 & 2 - 2 \cdot 2 & 0 - 2 \cdot 0 & 6 - 2 \cdot 9 \end{array}]$

Step 4.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 0 & 9 \\ 0 & - 2 & 0 & - 12 \end{array}]$

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 $1$ .

$[\begin{array}{c} 1 & 2 & 0 & 9 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 12 \end{array}]$

Step 4.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 0 & 9 \\ 0 & 1 & 0 & 6 \end{array}]$

Step 4.3

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

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Step 4.3.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 9 - 2 \cdot 6 \\ 0 & 1 & 0 & 6 \end{array}]$

Step 4.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 3 \\ 0 & 1 & 0 & 6 \end{array}]$

Step 5

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

$x = 0$

$y = 6$

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