Matematika Berhingga Contoh

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

Langkah 1

Write the system as a matrix.

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

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 2.1

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

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Langkah 2.1.1

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

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

Langkah 2.1.2

Sederhanakan $R_{2}$ .

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

Langkah 2.2

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

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Langkah 2.2.1

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

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

Langkah 2.2.2

Sederhanakan $R_{2}$ .

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

Langkah 2.3

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

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Langkah 2.3.1

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

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

Langkah 2.3.2

Sederhanakan $R_{1}$ .

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

Langkah 3

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

$x = \frac{8}{3}$

$y = \frac{1}{3}$

Langkah 4

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

$(\frac{8}{3}, \frac{1}{3})$