When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, -x + y + 5z = 2, y + z = 0

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When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x &#8211; 2y = 5, -x + y + 5z = 2, y + z = 0

Suresh Chand · Accepted Answer

Solution: Given x - 2y = 5   ..............(i) -x + y + 5z = 2 ..............(ii) y + z = 0 ....................(iii) The given equations can be written in the form Ax = b Where A = (begin{bmatrix}1 & -2 & 0-1 & 1 & 5 0 & 1 & 1end{bmatrix}), x = (begin{bmatrix}x y zend{bmatrix}), b = (begin{bmatrix}5 2 0end{bmatrix}) Its augmented form is [A:b] ~ [A:b] ~ (begin{bmatrix}1 & -2 & 0 & : & 5-1 & 1 & 5 & : & 2 0 & 1 & 1 & : & 0end{bmatrix}) R2 → R2 + R1 ~ (begin{bmatrix}1 & -2 & 0 & : & 50 & -1 & 5 & : & 7 0 & 1 & 1 & : & 0end{bmatrix}) R2 → (-1) R2 ~ (begin{bmatrix}1 & -2 & 0 & : & 50 & 1 & -5 & : & -7 0 & 1 & 1 & : & 0end{bmatrix}) R1 → R1 + 2R2 , R3 → R3 - R2 ~ (begin{bmatrix}1 & 0 & -10 & : & -90 & 1 & -5 & : & -7 0 & 0 & 6 & : & 7end{bmatrix}) (R_3 to frac{1}{6} R_3) ~ (begin{bmatrix}1 & 0 & -10 & : & -90 & 1 & -5 & : & -7 0 & 0 & 1 & : & frac{7}{6}end{bmatrix}) R1 → R1 + 10R3 ,  R2 → R2 + 5R3 ~ (begin{bmatrix}1 & 0 & 0 & : & frac{8}{3}0 & 1 & 0 & : & frac{-7}{6} 0 & 0 & 1 & : & frac{7}{6}end{bmatrix}) From the above, We can say that x = (begin{bmatrix}x y zend{bmatrix}) = (begin{bmatrix}frac{8}{3} frac{-7}{6} frac{7}{6}end{bmatrix}) This is the required solution.

When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, -x + y + 5z = 2, y + z = 0

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When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, -x + y + 5z = 2, y + z = 0

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