{l}{x+y-3z+6=2c}{3x-y+z-t=2a}{-x+3y-z+t=2b} କୁ ସମାଧାନ କରନ୍ତୁ

$$\left\{ \begin{array}{l}{ x + y - 3 z + 6 = 2 c }\\{ 3 x - y + z - t = 2 a }\\{ - x + 3 y - z + t = 2 b }\end{array} \right.$$

x, y, z ପାଇଁ ସମାଧାନ କରନ୍ତୁ

$$x=\frac{b}{6}+\frac{c}{6}+\frac{t}{4}+\frac{2a}{3}-\frac{1}{2}$$

$$y=\frac{a}{3}+\frac{5b}{6}-\frac{c}{6}-\frac{t}{4}+\frac{1}{2}$$

$$z=\frac{a}{3}+\frac{b}{3}-\frac{2c}{3}+2$$

ପ୍ରତିବଦଳ ବ୍ୟବହାର କରିବା ପଦକ୍ଷେପଗୁଡିକ

x ପାଇଁ x+y-3z+6=2c ସମାଧାନ କରନ୍ତୁ.

$$x=-y+3z-6+2c$$

ଦ୍ୱିତୀୟ ଏବଂ ତୃତୀୟ ସମୀକରଣରେ x ସ୍ଥାନରେ -y+3z-6+2c ପ୍ରତିବଦଳ କରନ୍ତୁ.

$$3\left(-y+3z-6+2c\right)-y+z-t=2a -\left(-y+3z-6+2c\right)+3y-z+t=2b$$

ଯଥାକ୍ରମେ y ଏବଂ z ପାଇଁ ଏହି ସମୀକରଣ ସମାଧାନ କରନ୍ତୁ.

$$y=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a z=y-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t$$

ସମୀକରଣ z=y-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t ରେ y ସ୍ଥାନରେ \frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a ପ୍ରତିବଦଳ କରନ୍ତୁ.

$$z=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t$$

z ପାଇଁ z=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t ସମାଧାନ କରନ୍ତୁ.

$$z=2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b$$

ସମୀକରଣ y=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a ରେ z ସ୍ଥାନରେ 2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b ପ୍ରତିବଦଳ କରନ୍ତୁ.

$$y=\frac{5}{2}\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a$$

y=\frac{5}{2}\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a ରୁ y ଗଣନା କରନ୍ତୁ.

$$y=\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b$$

ସମୀକରଣx=-y+3z-6+2c ରେ z ସ୍ଥାନରେ y ଏବଂ 2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b ପାଇଁ \frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b ପ୍ରତିବଦଳ କରନ୍ତୁ.

$$x=-\left(\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b\right)+3\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-6+2c$$

x=-\left(\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b\right)+3\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-6+2c ରୁ x ଗଣନା କରନ୍ତୁ.

$$x=-\frac{1}{2}+\frac{1}{6}c+\frac{1}{4}t+\frac{2}{3}a+\frac{1}{6}b$$

ବର୍ତ୍ତମାନ ସିଷ୍ଟମ୍‌ ସମାଧାନ ହୋଇଛି.

$$x=-\frac{1}{2}+\frac{1}{6}c+\frac{1}{4}t+\frac{2}{3}a+\frac{1}{6}b y=\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b z=2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b$$