Question

solve the equations 2x-y+3z=1, x+2y-z=2, 5y-5z=3 using matrix method
do put a pic of your solution!

Answer 1

Matrix Method

Solution.

The given system of linear equations is

$\color{red}{\begin{array}{cccccccc}2x&-&y&+&3z&=&1&\quad...(1)\\ x&+&2y&-&z&=&2&\quad...(2)\\ 0.x&+&5y&-&5z&=&3&\quad...(3)\end{array}}$

The system of equations can be put in matrix form,

$\quad\quad AX=B$,

where $A=\left[\begin{array}{ccc}2&-1&3\\1&2&-1\\0&5&-5\end{array}\right]$,

$X=\left[\begin{array}{c}x\\y\\z\end{array}\right]$ and $B=\left[\begin{array}{c}1\\2\\3\end{array}\right]$

Now, $|A|=\left|\begin{array}{ccc}2&-1&3\\1&2&-1\\0&5&-5\end{array}\right|$

$\quad=2(-10+5)+1(-5-0)+3(5-0)$

$\quad$( Expanding along $R_{1}$ )

$\quad=2(-5)+1(-5)+3(5)$

$\quad=-10-5+15$

$\quad=0$

Now, $Adj\:A=\left[\begin{array}{ccc}-5&5&5\\10&-10&-10\\-5&5&5\end{array}\right]^{T}$

$=\left[\begin{array}{ccc}-5&10&-5\\5&-10&5\\5&-10&5\end{array}\right]$

$\therefore (Adj\:A)B=\left[\begin{array}{ccc}-5&10&-5\\5&-10&5\\5&-10&5\end{array}\right]\left[\begin{array}{c}1\\2\\3\end{array}\right]$

$=\left[\begin{array}{c}-5+20-15\\5-20+15\\5-20+15\end{array}\right]$

$=\left[\begin{array}{c}0\\0\\0\end{array}\right]$

$=O$

$\Rightarrow \color{blue}{(Adj\:A)B=O}$

Thus the given system of equations can have either no solution or infinitely many solutions.

Note. To find any solution, put $k$, a numerical value in place of any of $x$ or $y$ or $z$. The system will reduce to a system of linear equations in two variables. Solve them to find a solution of the form $(f_{1}(k),\:f_{2}(k),\:k)$ when we take $z=k$.

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A shop sells three commodities. X purchases 2 units of q and sells 3 units of p and 3 units r. Y purchases 1 units of r and sells 2 units of p and 1 unit of q. Z purchases 3 unit of q and sells 4 units of p and 2 units of r. In the process X, Y, and Z earns Rs. 8,000, Rs.1000, and Rs. 4,000 respectively when price of item is p, q, and r per unit respectively (Consider selling units is positive earning and buying the units negative earnings) i) Write system of simultaneous equations ii) Find det(A) iii) Write transpose of A. Using matrices, find prices per unit of the three commodities.

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