Question

solve the eqution x-y+z=1,2x-y=1,3x+3y-4z=2 by reduction method.

Answer 1

2x-y=1
x=(y+1)/2. ..........(1)

x-y+z=1
[(y+1)/2]-y+z=1
-y+2z=1. ..........(2)
-9y+18z=9

3×{(y+1)/2}+3y-4z=2
9y-8z=1

-9y+18z=9
+9y-8z=1
_________
10z=10
z=1


add the value of z into ...(2)..

-y+2z=1
y=1

and x=1

Answer 2

Given :

x – y + z = 1,  2x – y = 1, 3x + 3y – 4z = 2.

To Find : Solve by matrix inverse method :

Solution:

x – y + z = 1,

2x – y  + 0.z = 1

3x + 3y – 4z = 2

AX = B

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

$B = \left[\begin{array}{ccc}1 \\1\\2\end{array}\right]$

X  =  A⁻¹B

| A|  =   5

A⁻¹ = Adj A/ | A |

$A^{-1}=\dfrac{1}{5} \left[\begin{array}{ccc}4&-1&1\\8&-7&2\\9&-6&1\end{array}\right]$

  $X=\dfrac{1}{5} \left[\begin{array}{ccc}5\\5\\5\end{array}\right]$

$X = \left[\begin{array}{ccc}1\\1\\1\end{array}\right]$

x = 1 , y = 1 , z = 1

x-y+z=1

2x-y=1

=> x - z  = 0  => x = z

=>  3x+3y-4z=2  => 3y - x  =  2

3y - x  =  2 => 6y - 2x = 4

2x-y=1

Adding both 5y  = 5  => y = 1

=> x = 1

Hence  z = 1

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