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Sagot :
The solution for the given system of equation is (x,y,z) is = (0 , (3-4t) , t) .
In the question ,
it is given that the system of equations are ,
3x + 3y + 12z = 9
x + y + 4z = 3
2x + 5y + 20z = 15
-x + 2y + 8z = 6
we have to solve the equations for x, y , z.
writing the given equations in augmented matrix form ,
we get ,
[tex]\left[\begin{array}{ccc}3&3&12\\1&1&4\\2&5&20\\-1&2&8\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}9\\3\\15\\6\end{array}\right][/tex]
Applying the operation , R₁/3 ⇒ R₁
[tex]\left[\begin{array}{ccc}1&1&4\\1&1&4\\2&5&20\\-1&2&8\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}3\\3\\15\\6\end{array}\right][/tex]
Applying the row operation , R₂ - R₁ ⇒ R₂ .
[tex]\left[\begin{array}{ccc}1&1&4\\0&0&0\\2&5&20\\-1&2&8\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}3\\0\\15\\6\end{array}\right][/tex]
After applying more row operations , in the above matrix ,
to make the lower triangle matrix 0 ,
we get ,
[tex]\left[\begin{array}{ccc}1&0&0\\0&1&4\\0&0&0\\0&0&0\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}0\\3\\0\\0\end{array}\right][/tex]
The above result can be expressed as x = 0,
and y + 4z = 3 .
So , the system of equations have infinite solutions .
So , the solution x, y, and z expressed in terms of the parameter t is
(0 , (3-4t) , t) .
Therefore , the solution is (x,y,z) is = (0 , (3-4t) , t) .
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