Matematika Berhingga Contoh

$6 x - 5 y = - 3$ , $8 x - y = 13$

Langkah 1

Write the system as a matrix.

$[\begin{array}{c} 6 & - 5 & - 3 \\ 8 & - 1 & 13 \end{array}]$

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 2.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Langkah 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6}{6} & \frac{- 5}{6} & \frac{- 3}{6} \\ 8 & - 1 & 13 \end{array}]$

Langkah 2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{5}{6} & - \frac{1}{2} \\ 8 & - 1 & 13 \end{array}]$

Langkah 2.2

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Langkah 2.2.1

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{6} & - \frac{1}{2} \\ 8 - 8 \cdot 1 & - 1 - 8 (- \frac{5}{6}) & 13 - 8 (- \frac{1}{2}) \end{array}]$

Langkah 2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{6} & - \frac{1}{2} \\ 0 & \frac{17}{3} & 17 \end{array}]$

Langkah 2.3

Multiply each element of $R_{2}$ by $\frac{3}{17}$ to make the entry at $2, 2$ a $1$ .

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Langkah 2.3.1

Multiply each element of $R_{2}$ by $\frac{3}{17}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{5}{6} & - \frac{1}{2} \\ \frac{3}{17} \cdot 0 & \frac{3}{17} \cdot \frac{17}{3} & \frac{3}{17} \cdot 17 \end{array}]$

Langkah 2.3.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{6} & - \frac{1}{2} \\ 0 & 1 & 3 \end{array}]$

Langkah 2.4

Perform the row operation $R_{1} = R_{1} + \frac{5}{6} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Langkah 2.4.1

Perform the row operation $R_{1} = R_{1} + \frac{5}{6} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{5}{6} \cdot 0 & - \frac{5}{6} + \frac{5}{6} \cdot 1 & - \frac{1}{2} + \frac{5}{6} \cdot 3 \\ 0 & 1 & 3 \end{array}]$

Langkah 2.4.2

Sederhanakan $R_{1}$ .

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

Langkah 3

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

$x = 2$

$y = 3$

Langkah 4

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

$(2, 3)$