फाइनाइट मैथ उदाहरण

$9 x - 6 y + 4 z = - 9$ , $5 x - 4 y + 9 z = 1$ , $3 y - 3 z = - 6$

चरण 1

Write the system as a matrix.

$[\begin{array}{c} 9 & - 6 & 4 & - 9 \\ 5 & - 4 & 9 & 1 \\ 0 & 3 & - 3 & - 6 \end{array}]$

चरण 2

घटी हुई पंक्ति के सोपानक रूप का पता करें.

और स्टेप्स के लिए टैप करें…

चरण 2.1

Multiply each element of $R_{1}$ by $\frac{1}{9}$ to make the entry at $1, 1$ a $1$ .

और स्टेप्स के लिए टैप करें…

चरण 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{9}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{9}{9} & \frac{- 6}{9} & \frac{4}{9} & \frac{- 9}{9} \\ 5 & - 4 & 9 & 1 \\ 0 & 3 & - 3 & - 6 \end{array}]$

चरण 2.1.2

$R_{1}$ को सरल करें.

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 5 & - 4 & 9 & 1 \\ 0 & 3 & - 3 & - 6 \end{array}]$

चरण 2.2

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

और स्टेप्स के लिए टैप करें…

चरण 2.2.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 5 - 5 \cdot 1 & - 4 - 5 (- \frac{2}{3}) & 9 - 5 (\frac{4}{9}) & 1 - 5 \cdot - 1 \\ 0 & 3 & - 3 & - 6 \end{array}]$

चरण 2.2.2

$R_{2}$ को सरल करें.

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 & - \frac{2}{3} & \frac{61}{9} & 6 \\ 0 & 3 & - 3 & - 6 \end{array}]$

चरण 2.3

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

और स्टेप्स के लिए टैप करें…

चरण 2.3.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ - \frac{3}{2} \cdot 0 & - \frac{3}{2} (- \frac{2}{3}) & - \frac{3}{2} \cdot \frac{61}{9} & - \frac{3}{2} \cdot 6 \\ 0 & 3 & - 3 & - 6 \end{array}]$

चरण 2.3.2

$R_{2}$ को सरल करें.

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 & 1 & - \frac{61}{6} & - 9 \\ 0 & 3 & - 3 & - 6 \end{array}]$

चरण 2.4

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

और स्टेप्स के लिए टैप करें…

चरण 2.4.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 & 1 & - \frac{61}{6} & - 9 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & - 3 - 3 (- \frac{61}{6}) & - 6 - 3 \cdot - 9 \end{array}]$

चरण 2.4.2

$R_{3}$ को सरल करें.

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 & 1 & - \frac{61}{6} & - 9 \\ 0 & 0 & \frac{55}{2} & 21 \end{array}]$

चरण 2.5

Multiply each element of $R_{3}$ by $\frac{2}{55}$ to make the entry at $3, 3$ a $1$ .

और स्टेप्स के लिए टैप करें…

चरण 2.5.1

Multiply each element of $R_{3}$ by $\frac{2}{55}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 & 1 & - \frac{61}{6} & - 9 \\ \frac{2}{55} \cdot 0 & \frac{2}{55} \cdot 0 & \frac{2}{55} \cdot \frac{55}{2} & \frac{2}{55} \cdot 21 \end{array}]$

चरण 2.5.2

$R_{3}$ को सरल करें.

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 & 1 & - \frac{61}{6} & - 9 \\ 0 & 0 & 1 & \frac{42}{55} \end{array}]$

चरण 2.6

Perform the row operation $R_{2} = R_{2} + \frac{61}{6} R_{3}$ to make the entry at $2, 3$ a $0$ .

और स्टेप्स के लिए टैप करें…

चरण 2.6.1

Perform the row operation $R_{2} = R_{2} + \frac{61}{6} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 + \frac{61}{6} \cdot 0 & 1 + \frac{61}{6} \cdot 0 & - \frac{61}{6} + \frac{61}{6} \cdot 1 & - 9 + \frac{61}{6} \cdot \frac{42}{55} \\ 0 & 0 & 1 & \frac{42}{55} \end{array}]$

चरण 2.6.2

$R_{2}$ को सरल करें.

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{9} & - 1 \\ 0 & 1 & 0 & - \frac{68}{55} \\ 0 & 0 & 1 & \frac{42}{55} \end{array}]$

चरण 2.7

Perform the row operation $R_{1} = R_{1} - \frac{4}{9} R_{3}$ to make the entry at $1, 3$ a $0$ .

और स्टेप्स के लिए टैप करें…

चरण 2.7.1

Perform the row operation $R_{1} = R_{1} - \frac{4}{9} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{4}{9} \cdot 0 & - \frac{2}{3} - \frac{4}{9} \cdot 0 & \frac{4}{9} - \frac{4}{9} \cdot 1 & - 1 - \frac{4}{9} \cdot \frac{42}{55} \\ 0 & 1 & 0 & - \frac{68}{55} \\ 0 & 0 & 1 & \frac{42}{55} \end{array}]$

चरण 2.7.2

$R_{1}$ को सरल करें.

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 & - \frac{221}{165} \\ 0 & 1 & 0 & - \frac{68}{55} \\ 0 & 0 & 1 & \frac{42}{55} \end{array}]$

चरण 2.8

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

और स्टेप्स के लिए टैप करें…

चरण 2.8.1

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

$[\begin{array}{c} 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 0 + \frac{2}{3} \cdot 0 & - \frac{221}{165} + \frac{2}{3} (- \frac{68}{55}) \\ 0 & 1 & 0 & - \frac{68}{55} \\ 0 & 0 & 1 & \frac{42}{55} \end{array}]$

चरण 2.8.2

$R_{1}$ को सरल करें.

$[\begin{array}{c} 1 & 0 & 0 & - \frac{119}{55} \\ 0 & 1 & 0 & - \frac{68}{55} \\ 0 & 0 & 1 & \frac{42}{55} \end{array}]$

चरण 3

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

$x = - \frac{119}{55}$

$y = - \frac{68}{55}$

$z = \frac{42}{55}$

चरण 4

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

$(- \frac{119}{55}, - \frac{68}{55}, \frac{42}{55})$