Pré-cálculo Exemplos

2 x + 4 y = 25

$2 x + 4 y = 25$ ,

x + 5 y = 2

$x + 5 y = 2$

Etapa 1

Encontre

A X = B

$A X = B$ do sistema de equações.

[\begin{array}{c} 2 & 4 \\ 1 & 5 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 25 \\ 2 \end{array}]

$[\begin{array}{c} 2 & 4 \\ 1 & 5 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 25 \\ 2 \end{array}]$

Etapa 2

Encontre o inverso da matriz do coeficiente.

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Etapa 2.1

The inverse of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]

$\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where

a d - b c

$a d - b c$ is the determinant.

Etapa 2.2

Find the determinant.

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Etapa 2.2.1

O determinante de uma matriz

2 \times 2

$2 \times 2$ pode ser encontrado ao usar a fórmula

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

2 \cdot 5 - 1 \cdot 4

$2 \cdot 5 - 1 \cdot 4$

Etapa 2.2.2

Simplifique o determinante.

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Etapa 2.2.2.1

Simplifique cada termo.

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Etapa 2.2.2.1.1

Multiplique

2

$2$ por

5

$5$ .

10 - 1 \cdot 4

$10 - 1 \cdot 4$

Etapa 2.2.2.1.2

Multiplique

- 1

$- 1$ por

4

$4$ .

10 - 4

$10 - 4$

10 - 4

$10 - 4$

Etapa 2.2.2.2

Subtraia

4

$4$ de

10

$10$ .

6

$6$

6

$6$

6

$6$

Etapa 2.3

Since the determinant is non-zero, the inverse exists.

Etapa 2.4

Substitute the known values into the formula for the inverse.

\frac{1}{6} [\begin{array}{c} 5 & - 4 \\ - 1 & 2 \end{array}]

$\frac{1}{6} [\begin{array}{c} 5 & - 4 \\ - 1 & 2 \end{array}]$

Etapa 2.5

Multiplique

\frac{1}{6}

$\frac{1}{6}$ por cada elemento da matriz.

[\begin{array}{c} \frac{1}{6} \cdot 5 & \frac{1}{6} \cdot - 4 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{6} \cdot 5 & \frac{1}{6} \cdot - 4 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6

Simplifique cada elemento da matriz.

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Etapa 2.6.1

Combine

\frac{1}{6}

$\frac{1}{6}$ e

5

$5$ .

[\begin{array}{c} \frac{5}{6} & \frac{1}{6} \cdot - 4 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & \frac{1}{6} \cdot - 4 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.2

Cancele o fator comum de

2

$2$ .

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Etapa 2.6.2.1

Fatore

2

$2$ de

6

$6$ .

[\begin{array}{c} \frac{5}{6} & \frac{1}{2 (3)} \cdot - 4 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & \frac{1}{2 (3)} \cdot - 4 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.2.2

Fatore

2

$2$ de

- 4

$- 4$ .

[\begin{array}{c} \frac{5}{6} & \frac{1}{2 \cdot 3} \cdot (2 \cdot - 2) \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & \frac{1}{2 \cdot 3} \cdot (2 \cdot - 2) \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.2.3

Cancele o fator comum.

[\begin{array}{c} \frac{5}{6} & \frac{1}{2 \cdot 3} \cdot (2 \cdot - 2) \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & \frac{1}{2 \cdot 3} \cdot (2 \cdot - 2) \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.2.4

Reescreva a expressão.

[\begin{array}{c} \frac{5}{6} & \frac{1}{3} \cdot - 2 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & \frac{1}{3} \cdot - 2 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

[\begin{array}{c} \frac{5}{6} & \frac{1}{3} \cdot - 2 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & \frac{1}{3} \cdot - 2 \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.3

Combine

\frac{1}{3}

$\frac{1}{3}$ e

- 2

$- 2$ .

[\begin{array}{c} \frac{5}{6} & \frac{- 2}{3} \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & \frac{- 2}{3} \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.4

Mova o número negativo para a frente da fração.

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ \frac{1}{6} \cdot - 1 & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.5

Combine

\frac{1}{6}

$\frac{1}{6}$ e

- 1

$- 1$ .

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ \frac{- 1}{6} & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ \frac{- 1}{6} & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.6

Mova o número negativo para a frente da fração.

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{6} \cdot 2 \end{array}]$

Etapa 2.6.7

Cancele o fator comum de

2

$2$ .

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Etapa 2.6.7.1

Fatore

2

$2$ de

6

$6$ .

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{2 (3)} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{2 (3)} \cdot 2 \end{array}]$

Etapa 2.6.7.2

Cancele o fator comum.

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{2 \cdot 3} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{2 \cdot 3} \cdot 2 \end{array}]$

Etapa 2.6.7.3

Reescreva a expressão.

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]$

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]$

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]$

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}]$

Etapa 3

Multiplique pela esquerda os dois lados da equação da matriz pela matriz inversa.

([\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] \cdot [\begin{array}{c} 2 & 4 \\ 1 & 5 \end{array}]) \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] \cdot [\begin{array}{c} 25 \\ 2 \end{array}]

$([\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] \cdot [\begin{array}{c} 2 & 4 \\ 1 & 5 \end{array}]) \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] \cdot [\begin{array}{c} 25 \\ 2 \end{array}]$

Etapa 4

Qualquer matriz multiplicada por seu inverso é sempre igual a

1

$1$ .

A \cdot A^{- 1} = 1

$A \cdot A^{- 1} = 1$ .

[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] \cdot [\begin{array}{c} 25 \\ 2 \end{array}]

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] \cdot [\begin{array}{c} 25 \\ 2 \end{array}]$

Etapa 5

Multiplique

[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] [\begin{array}{c} 25 \\ 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} & - \frac{2}{3} \\ - \frac{1}{6} & \frac{1}{3} \end{array}] [\begin{array}{c} 25 \\ 2 \end{array}]$ .

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Etapa 5.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is

2 \times 2

$2 \times 2$ and the second matrix is

2 \times 1

$2 \times 1$ .

Etapa 5.2

Multiplique cada linha na primeira matriz por cada coluna na segunda matriz.

[\begin{array}{c} \frac{5}{6} \cdot 25 - \frac{2}{3} \cdot 2 \\ - \frac{1}{6} \cdot 25 + \frac{1}{3} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{5}{6} \cdot 25 - \frac{2}{3} \cdot 2 \\ - \frac{1}{6} \cdot 25 + \frac{1}{3} \cdot 2 \end{array}]$

Etapa 5.3

Simplifique cada elemento da matriz multiplicando todas as expressões.

[\begin{array}{c} \frac{39}{2} \\ - \frac{7}{2} \end{array}]

$[\begin{array}{c} \frac{39}{2} \\ - \frac{7}{2} \end{array}]$

[\begin{array}{c} \frac{39}{2} \\ - \frac{7}{2} \end{array}]

$[\begin{array}{c} \frac{39}{2} \\ - \frac{7}{2} \end{array}]$

Etapa 6

Simplifique os lados esquerdo e direito.

[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{39}{2} \\ - \frac{7}{2} \end{array}]

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{39}{2} \\ - \frac{7}{2} \end{array}]$

Etapa 7

Encontre a solução.

x = \frac{39}{2}

$x = \frac{39}{2}$

y = - \frac{7}{2}

$y = - \frac{7}{2}$

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