Álgebra Exemplos

3 x + 3 y = 3

$3 x + 3 y = 3$ ,

x < y

$x < y$

Etapa 1

Introduza as variáveis de folga

u

$u$ e

v

$v$ para substituir as desigualdades por equações.

x + Z = y

$x + Z = y$

3 x + 3 y - 3 = 0

$3 x + 3 y - 3 = 0$

Etapa 2

Subtraia

y

$y$ dos dois lados da equação.

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

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

Etapa 3

Some

3

$3$ aos dois lados da equação.

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

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

Etapa 4

Escreva o sistema de equações em formato de matriz.

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

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

Etapa 5

Encontre a forma escalonada reduzida por linhas.

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Etapa 5.1

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Etapa 5.1.1

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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

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

Etapa 5.1.2

Simplifique

R_{2}

$R_{2}$ .

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

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

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

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

Etapa 5.2

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{6}

$\frac{1}{6}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Etapa 5.2.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{6}

$\frac{1}{6}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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

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

Etapa 5.2.2

Simplifique

R_{2}

$R_{2}$ .

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

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

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

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

Etapa 5.3

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

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Etapa 5.3.1

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

[\begin{matrix} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + \frac{1}{2} 0 & 1 & 0 & \frac{1}{2} \end{matrix}]

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

Etapa 5.3.2

Simplifique

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Etapa 6

Use a matriz de resultados para declarar as soluções finais ao sistema de equações.

x = \frac{1}{2}

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

y = \frac{1}{2}

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

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