Question

Solving system of equation X-2Y=10, 2X-Y-Z=8, -2Y+Z=7 using matrix method then value of x,y,z are given by

Answer 1

Given : system of equation X-2Y=10, 2X-Y-Z=8, -2Y+Z=7

To Find : Solve for X , Y  & Z

Solution:

X  - 2Y  + 0.Z = 10   Eq1

2X - Y  - Z  = 8    Eq2

0.X - 2Y +  Z  =  7    Eq3

Eq1 - Eq3

=>  X - Z = 3

2* Eq2 - Eq1

=> 3X - 2Z  = 6

Z = - 3   , X = 0   , Y = -5

Matrix Method

$\left[\begin{array}{ccc}1&-2&0\\2&-1&-1\\0&-2&1\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}10\\8\\7\end{array}\right]$

AX = B

=> X = A⁻¹B

| A | = 1   Hence inverse Exist

$A^{-1} = \left[\begin{array}{ccc}-3& 2&2\\-2& 1& 1\\-4& 2&3\end{array}\right]$

$\left[\begin{array}{ccc}-3& 2&2\\-2& 1&1\\-4&2&3\end{array}\right]\left[\begin{array}{ccc}10\\8\\7\end{array}\right] = \left[\begin{array}{ccc} 0\\-5\\-3\end{array}\right]$

X = 0   , Y = -5  , Z = - 3

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Answer 2

$\sf{Given\::\:\:\begin{bmatrix}\sf{x-2y=10}\\ \sf{2x-y-z=8}\\\sf{ -2y+z=7}\end{bmatrix}}$

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$\sf{x-2y=10}$

$\implies\sf{x-2y+2y=10+2y}$

$\implies\sf{x=10+2y}$

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$\bf{Subsititute\:x=10+2y}$

$\sf{\begin{bmatrix}\sf{-2y+z=7}\\\sf{ 2\left(10+2y\right)-y-z=8}\end{bmatrix}}$

$\begin{bmatrix}\sf{-2y+z=7}\\ \sf{20+3y-z=8}\end{bmatrix}$

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$\bf{\bf{Subsititute\:}y=-\dfrac{7-z}{2}}$

$\begin{bmatrix}\sf{20+3\left(-\dfrac{7-z}{2}\right)-z=8}\end{bmatrix}$

$\begin{bmatrix}\sf{\dfrac{-21+z}{2}+20=8}\end{bmatrix}$

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$\sf{\dfrac{-21+z}{2}+20=8}$

$\implies\sf{\dfrac{-21+z}{2}+20-20=8-20}$

$\implies\sf{\dfrac{-21+z}{2}=-12}$

$\implies\sf{\dfrac{2\left(-21+z\right)}{2}=2\left(-12\right)}$

$\implies\sf{-21+z=-24}$

$\implies\sf{-21+z+21=-24+21}$

$\implies\sf{z=-3}$

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$\bf{\bf{For\:}y=-\dfrac{7-z}{2}}$

$\bf{\bf{Subsititute\:}z=-3}$

$\sf{y=-\dfrac{7-\left(-3\right)}{2}}$

$\implies\sf{y=-\dfrac{7+3}{2}}$

$\implies\sf{y=-\dfrac{10}{2}}$

$\implies\sf{y=5}$

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$\bf{For\:x=10+2y}$

$\bf{\bf{Subsititute\:}z=-3,\:y=-5}$

$\sf{x=10+2\left(-5\right)}$

$\implies\sf{x=10-10}$

$\implies\sf{x=0}$

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$\sf{The\:solutions\:to\:the\:system\:of\:equations\:are:}$

$\boxed{\boxed{\bf{x=0,\:z=-3,\:y=-5}}}$