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What values of x, y and z satisfy the following system of linear equations?

$$\begin{bmatrix} 1 &2 &3 \\ 1& 3 &4 \\ 2& 2 &3 \end{bmatrix} \begin{bmatrix} x\\y \\ z \end{bmatrix} = \begin{bmatrix} 6\\8 \\ 12 \end{bmatrix}$$

- $x = 6$, $y = 3$, $z = 2$
- $x = 12$, $y = 3$, $z = - 4$
- $x = 6$, $y = 6$, $z = - 4$
- $x = 12$, $y = - 3$, $z = 0$

18 votes

Best answer

$\begin{bmatrix} 1 &2 &3 \\ 1& 3 &4 \\ 2& 2 &3 \end{bmatrix} \begin{bmatrix} x\\y \\ z \end{bmatrix} = \begin{bmatrix} 6\\8 \\ 12 \end{bmatrix}$

- $x+2y+3z=6\quad \to (1)$
- $x+3y+4z=8\quad \to(2)$
- $2x+2y+3z=12\quad \to (3)$

Apply $(3)-(1)$

$\implies x=6$

Put value of $x$ in any of the $2$ equations. Let's take (1) and (2)

- $2y+3z=0$
- $3y +4z=2$
- $\implies y=6 , z=-4$

Hence, Option(C) $x=6,y=6,z=-4$.

19 votes

I will tell bit faster method to solve such questions

System is

$\begin{bmatrix} 1 & 2&3 \\ 1 & 3 &4 \\ 2& 2 & 3 \end{bmatrix}$ $\begin{bmatrix} x\\ y\\ z \end{bmatrix}$ = $\begin{bmatrix} 6\\ 8\\ 12 \end{bmatrix}$

So, what question asks is what combination of (X*column1) + (Y*column2) +(Z*column3) produces the vector on the right-hand side.

$X\begin{bmatrix} 1\\ 1\\ 2 \end{bmatrix}$ + $Y\begin{bmatrix} 2\\ 3\\ 2 \end{bmatrix}$ + $Z\begin{bmatrix} 3\\ 4\\ 3 \end{bmatrix}$ =$\begin{bmatrix} 6\\ 8\\ 12 \end{bmatrix}$

(A)X=6, Y=3, Z=2

LHS comes to be while evaluating

$\begin{bmatrix} {\color{Red} 6} &{\color{Red} 6} &{\color{Red} 6} \\ & & \\ & & \end{bmatrix}$

The first column will not evaluate to be 6 on RHS(6+6+6 is not equal to 6), so we stop here only and reject option (A).

Option (B) X=12, Y=3, Z=-4

$\begin{bmatrix} {\color{Green} 12} & {\color{Green} 6} & {\color{Green} -12}\\ {\color{Red} 12} &{\color{Red} 9} & {\color{Red} -16} \\ & & \end{bmatrix}$

The second entry of column2 does not add up to match 8 on the RHS so we stop our calculation here and reject this option too.

Option (c) X=6, Y=6, Z=-4

$\begin{bmatrix} 6 & 12 &-12 \\ 6 & 18 &-16 \\ 12 &12 & -12 \end{bmatrix}$ = $\begin{bmatrix} 6\\ 8\\ 12 \end{bmatrix}$

Yes this matches our answer. **Ans-(C)**

At first this method may look complicated but this is fastest method to solve such question.Not more than 50Secs. :P