Pré-cálculo Exemplos

$2 x + y = 5$ , $x - y = 1$

Etapa 1

Write the system as a matrix.

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

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Etapa 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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

Etapa 2.1.2

Simplifique $R_{1}$ .

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

Etapa 2.2

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

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Etapa 2.2.1

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

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

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{5}{2} \\ 0 & - \frac{3}{2} & - \frac{3}{2} \end{array}]$

Etapa 2.3

Multiply each element of $R_{2}$ by $- \frac{2}{3}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $- \frac{2}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{5}{2} \\ - \frac{2}{3} \cdot 0 & - \frac{2}{3} (- \frac{3}{2}) & - \frac{2}{3} (- \frac{3}{2}) \end{array}]$

Etapa 2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & 1 \end{array}]$

Etapa 2.4

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

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Etapa 2.4.1

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

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & \frac{5}{2} - \frac{1}{2} \cdot 1 \\ 0 & 1 & 1 \end{array}]$

Etapa 2.4.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = 2$

$y = 1$

Etapa 4

The solution is the set of ordered pairs that make the system true.

$(2, 1)$