Given:
{ 5 x + 6 y + 7 z = 40 2 x + 4 y + 2 z = 34 x + 3 y + 5 z = 30 \left\{
\begin{array}{ c c c }
5x+6y+7z & = & 40 \\
2x+4y+2z & = & 34\\
x+3y+5z & = & 30 \\
\end{array}\right. ⎩ ⎨ ⎧ 5 x + 6 y + 7 z 2 x + 4 y + 2 z x + 3 y + 5 z = = = 40 34 30
Solution:
CRAMMER'S RULE
det ∣ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ∣ = a 11 ∗ a 22 ∗ a 33 + a 12 ∗ a 23 ∗ a 31 + a 13 ∗ a 21 ∗ a 32 − − a 13 ∗ a 22 ∗ a 31 − a 11 ∗ a 23 ∗ a 32 − a 12 ∗ a 21 ∗ a 33 \det\left|
\begin{array}{ c c c }
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33} \\
\end{array}\right|=a_{11}*a_{22}*a_{33}+a_{12}*a_{23}*a_{31}+a_{13}*a_{21}*a_{32}-\\-a_{13}*a_{22}*a_{31}-a_{11}*a_{23}*a_{32}-a_{12}*a_{21}*a_{33} det ∣ ∣ a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 ∣ ∣ = a 11 ∗ a 22 ∗ a 33 + a 12 ∗ a 23 ∗ a 31 + a 13 ∗ a 21 ∗ a 32 − − a 13 ∗ a 22 ∗ a 31 − a 11 ∗ a 23 ∗ a 32 − a 12 ∗ a 21 ∗ a 33
x = D x D , y = D y D , z = D Z D x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_Z}{D} x = D D x , y = D D y , z = D D Z
D = det ∣ 5 6 7 2 4 2 1 3 5 ∣ = 5 ∗ 4 ∗ 5 + 6 ∗ 2 ∗ 1 + 7 ∗ 2 ∗ 3 − 7 ∗ 4 ∗ 1 − − 5 ∗ 2 ∗ 3 − 6 ∗ 2 ∗ 5 = 36 D=\det\left|
\begin{array}{ c c c }
5 & 6 & 7 \\
2 & 4 & 2\\
1 & 3 & 5 \\
\end{array}\right|=5*4*5+6*2*1+7*2*3-7*4*1-\\
-5*2*3-6*2*5=36 D = det ∣ ∣ 5 2 1 6 4 3 7 2 5 ∣ ∣ = 5 ∗ 4 ∗ 5 + 6 ∗ 2 ∗ 1 + 7 ∗ 2 ∗ 3 − 7 ∗ 4 ∗ 1 − − 5 ∗ 2 ∗ 3 − 6 ∗ 2 ∗ 5 = 36
D x = det ∣ 40 6 7 34 4 2 30 3 5 ∣ = 40 ∗ 4 ∗ 5 + 6 ∗ 2 ∗ 30 + 7 ∗ 34 ∗ 3 − 7 ∗ 4 ∗ 30 − − 40 ∗ 2 ∗ 3 − 6 ∗ 34 ∗ 5 = − 226 D_x=\det\left|
\begin{array}{ c c c }
40 & 6 & 7 \\
34 & 4 & 2\\
30 & 3 & 5 \\
\end{array}\right|=40*4*5+6*2*30+7*34*3-7*4*30-\\-40*2*3-6*34*5=-226 D x = det ∣ ∣ 40 34 30 6 4 3 7 2 5 ∣ ∣ = 40 ∗ 4 ∗ 5 + 6 ∗ 2 ∗ 30 + 7 ∗ 34 ∗ 3 − 7 ∗ 4 ∗ 30 − − 40 ∗ 2 ∗ 3 − 6 ∗ 34 ∗ 5 = − 226
D y = det ∣ 5 40 7 2 34 2 1 30 5 ∣ = 5 ∗ 34 ∗ 5 + 40 ∗ 2 ∗ 1 + 7 ∗ 2 ∗ 30 − 7 ∗ 34 ∗ 1 − − 5 ∗ 2 ∗ 30 − 40 ∗ 2 ∗ 5 = 412 D_y=\det\left|
\begin{array}{ c c c }
5 & 40 & 7 \\
2 & 34 & 2\\
1 & 30 & 5 \\
\end{array}\right|=5*34*5+40*2*1+7*2*30-7*34*1-\\-5*2*30-40*2*5=412 D y = det ∣ ∣ 5 2 1 40 34 30 7 2 5 ∣ ∣ = 5 ∗ 34 ∗ 5 + 40 ∗ 2 ∗ 1 + 7 ∗ 2 ∗ 30 − 7 ∗ 34 ∗ 1 − − 5 ∗ 2 ∗ 30 − 40 ∗ 2 ∗ 5 = 412
D z = det ∣ 5 6 40 2 4 34 1 3 30 ∣ = 5 ∗ 4 ∗ 30 + 6 ∗ 34 ∗ 1 + 40 ∗ 2 ∗ 3 − 40 ∗ 4 ∗ 1 − − 5 ∗ 34 ∗ 3 − 6 ∗ 2 ∗ 30 = 14 D_z=\det\left|
\begin{array}{ c c c }
5 & 6 & 40 \\
2 & 4 & 34\\
1 & 3 & 30 \\
\end{array}\right|=5*4*30+6*34*1+40*2*3-40*4*1-\\-5*34*3-6*2*30=14 D z = det ∣ ∣ 5 2 1 6 4 3 40 34 30 ∣ ∣ = 5 ∗ 4 ∗ 30 + 6 ∗ 34 ∗ 1 + 40 ∗ 2 ∗ 3 − 40 ∗ 4 ∗ 1 − − 5 ∗ 34 ∗ 3 − 6 ∗ 2 ∗ 30 = 14
x = D x D = − 226 36 = − 113 18 = − 6 5 18 ; y = D y D = 412 36 = 103 9 = 11 4 9 ; z = D z D = 14 36 = 7 18 x=\frac{D_x}{D}=\frac{-226}{36}=-\frac{113}{18}=-6\frac{5}{18};y=\frac{D_y}{D}=\frac{412}{36}=\frac{103}{9}=11\frac{4}{9};z=\frac{D_z}{D}=\frac{14}{36}=\frac{7}{18} x = D D x = 36 − 226 = − 18 113 = − 6 18 5 ; y = D D y = 36 412 = 9 103 = 11 9 4 ; z = D D z = 36 14 = 18 7
Answer :
x = − 6 5 18 ; y = 11 4 9 ; z = 7 18 x=-6\frac{5}{18};y=11\frac{4}{9};z=\frac{7}{18} x = − 6 18 5 ; y = 11 9 4 ; z = 18 7
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