Matematika Berhingga Contoh

$2 x - 3 y = 4$ , $2 x + 2 y = 9$

Langkah 1

Write the system as a matrix.

$[\begin{array}{c} 2 & - 3 & 4 \\ 2 & 2 & 9 \end{array}]$

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 2.1

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

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Langkah 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{- 3}{2} & \frac{4}{2} \\ 2 & 2 & 9 \end{array}]$

Langkah 2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & 2 \\ 2 & 2 & 9 \end{array}]$

Langkah 2.2

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

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Langkah 2.2.1

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

$[\begin{array}{c} 1 & - \frac{3}{2} & 2 \\ 2 - 2 \cdot 1 & 2 - 2 (- \frac{3}{2}) & 9 - 2 \cdot 2 \end{array}]$

Langkah 2.2.2

Sederhanakan $R_{2}$ .

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

Langkah 2.3

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Langkah 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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

Langkah 2.3.2

Sederhanakan $R_{2}$ .

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

Langkah 2.4

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

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Langkah 2.4.1

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

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

Langkah 2.4.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{7}{2} \\ 0 & 1 & 1 \end{array}]$

Langkah 3

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

$x = \frac{7}{2}$

$y = 1$

Langkah 4

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

$(\frac{7}{2}, 1)$