Algebra Examples

2 x + 2 y = 6

$2 x + 2 y = 6$ ,

x + 2 y > 9

$x + 2 y > 9$

Step 1

Introduce slack variables

u

$u$ and

v

$v$ to replace the inequalities with equations.

x + 2 y - Z = 9

$x + 2 y - Z = 9$

2 x + 2 y - 6 = 0

$2 x + 2 y - 6 = 0$

Step 2

Add

6

$6$ to both sides of the equation.

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

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

Step 3

Write the system of equations in matrix form.

[\begin{matrix} 1 & 2 & 0 & 9 2 & 2 & 0 & 6 \end{matrix}]

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

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}

$R_{2} = R_{2} - 2 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 4.1.1

Perform the row operation

R_{2} = R_{2} - 2 R_{1}

$R_{2} = R_{2} - 2 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{matrix} 1 & 2 & 0 & 9 2 - 2 \cdot 1 & 2 - 2 \cdot 2 & 0 - 2 \cdot 0 & 6 - 2 \cdot 9 \end{matrix}]

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

Step 4.1.2

Simplify

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & 2 & 0 & 9 0 & - 2 & 0 & - 12 \end{matrix}]

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

[\begin{matrix} 1 & 2 & 0 & 9 0 & - 2 & 0 & - 12 \end{matrix}]

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

Step 4.2

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 4.2.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{matrix} 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{matrix}]

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

Step 4.2.2

Simplify

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & 2 & 0 & 9 0 & 1 & 0 & 6 \end{matrix}]

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

[\begin{matrix} 1 & 2 & 0 & 9 0 & 1 & 0 & 6 \end{matrix}]

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

Step 4.3

Perform the row operation

R_{1} = R_{1} - 2 R_{2}

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Step 4.3.1

Perform the row operation

R_{1} = R_{1} - 2 R_{2}

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{matrix} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 9 - 2 \cdot 6 0 & 1 & 0 & 6 \end{matrix}]

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

Step 4.3.2

Simplify

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & 0 & 0 & - 3 0 & 1 & 0 & 6 \end{matrix}]

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

[\begin{matrix} 1 & 0 & 0 & - 3 0 & 1 & 0 & 6 \end{matrix}]

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

[\begin{matrix} 1 & 0 & 0 & - 3 0 & 1 & 0 & 6 \end{matrix}]

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

Step 5

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

$x = 0$

$y = 6$

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