Question #329268

x - 3y + 6z = 21

3x + 2y - 5z = - 30

2x - 5y + 2z = -6

The sum of values of x

x, y

y, and z

z of the solution is


1
Expert's answer
2022-04-18T01:48:24-0400

Let's solve the system of linear equations using the Cramer's rule.

Δ=136325252==122+(3)(5)2+63(5)6221(5)(5)(3)32==4+30902425+18=87;\Delta=\begin{vmatrix} 1& - 3&6\\ 3 & 2&-5\\ 2&-5&2 \end{vmatrix}=\\ =1\cdot2\cdot2+(-3)\cdot(-5)\cdot2+6\cdot3\cdot(-5)-\\-6\cdot2\cdot2-1\cdot(-5)\cdot(-5)-(-3)\cdot3\cdot2=\\ =4+30-90-24-25+18=-87;


Δ1=21363025652==2122+(3)(5)(6)+6(30)(5)62(6)21(5)(5)(3)(30)2==8490+900+72525180=261;\Delta_1=\begin{vmatrix} 21& - 3&6\\ -30 & 2&-5\\ -6&-5&2 \end{vmatrix}=\\ =21\cdot2\cdot2+(-3)\cdot(-5)\cdot(-6)+6\cdot(-30)\cdot(-5)-\\-6\cdot2\cdot(-6)-21\cdot(-5)\cdot(-5)-(-3)\cdot(-30)\cdot2=\\ =84-90+900+72-525-180=261;


Δ2=12163305262==1(30)2+21(5)2+63(6)6(30)21(5)(6)2132==60210108+36030126=174;\Delta_2=\begin{vmatrix} 1& 21&6\\ 3 & - 30&-5\\ 2&-6&2 \end{vmatrix}=\\ =1\cdot(-30)\cdot2+21\cdot(-5)\cdot2+6\cdot3\cdot(-6)-\\-6\cdot(-30)\cdot2-1\cdot(-5)\cdot(-6)-21\cdot3\cdot2=\\ =-60-210-108+360-30-126=-174;


Δ3=13213230256==12(6)+(3)(30)2+213(5)21221(30)(5)(3)3(6)==12+1803158415054=435;\Delta_3=\begin{vmatrix} 1& - 3&21\\ 3 & 2&-30\\ 2&-5&-6 \end{vmatrix}=\\ =1\cdot2\cdot(-6)+(-3)\cdot(-30)\cdot2+21\cdot3\cdot(-5)-\\-21\cdot2\cdot2-1\cdot(-30)\cdot(-5)-(-3)\cdot3\cdot(-6)=\\ =-12+180-315-84-150-54=-435;


x=Δ1Δ=261873;y=Δ2Δ=17487=2;z=Δ3Δ=43587=5;x+y+z=3+2+5=4.x=\cfrac{\Delta_1} {\Delta} =\cfrac{261}{-87}-3;\\ y=\cfrac{\Delta_2} {\Delta} =\cfrac{-174}{-87}=2;\\ z=\cfrac{\Delta_3} {\Delta} =\cfrac{-435}{-87}=5;\\ x+y+z=-3+2+5=4.






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