Matematica discreta Esempi

$3 x + 2 y + z = 3$ , $x - 2 y + z = 5$

Passaggio 1

Write the system as a matrix.

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

Passaggio 2

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 2.1

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

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Passaggio 2.1.1

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

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

Passaggio 2.1.2

Semplifica $R_{1}$ .

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

Passaggio 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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Passaggio 2.2.1

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

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

Passaggio 2.2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & 1 \\ 0 & - \frac{8}{3} & \frac{2}{3} & 4 \end{array}]$

Passaggio 2.3

Multiply each element of $R_{2}$ by $- \frac{3}{8}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 2.3.1

Multiply each element of $R_{2}$ by $- \frac{3}{8}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & 1 \\ - \frac{3}{8} \cdot 0 & - \frac{3}{8} (- \frac{8}{3}) & - \frac{3}{8} \cdot \frac{2}{3} & - \frac{3}{8} \cdot 4 \end{array}]$

Passaggio 2.3.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & 1 \\ 0 & 1 & - \frac{1}{4} & - \frac{3}{2} \end{array}]$

Passaggio 2.4

Perform the row operation $R_{1} = R_{1} - \frac{2}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Passaggio 2.4.1

Perform the row operation $R_{1} = R_{1} - \frac{2}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & \frac{1}{3} - \frac{2}{3} (- \frac{1}{4}) & 1 - \frac{2}{3} (- \frac{3}{2}) \\ 0 & 1 & - \frac{1}{4} & - \frac{3}{2} \end{array}]$

Passaggio 2.4.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{2} & 2 \\ 0 & 1 & - \frac{1}{4} & - \frac{3}{2} \end{array}]$

Passaggio 3

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

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

$y - \frac{1}{4} z = - \frac{3}{2}$

Passaggio 4

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

$(2 - \frac{z}{2}, - \frac{3}{2} + \frac{z}{4}, z)$