First slide

System of linear equations

Question

Let a, b and c be three real numbers satisfying
$[\begin{array}{lll} a b c \end{array}] [\begin{array}{lll} 1 9 7 \\ 8 2 7 \\ 7 3 7 \end{array}] = [\begin{array}{lll} 0 0 0 \end{array}]$ (1)
If the point $P (a, b, c)$ with reference to (1), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Moderate

A

0
B

12
C

7
D

6

Solution

We have
$a + 8 b + 7 c = 0, 9 a + 2 b + 3 c = 0, 7 a + 7 b + 7 c = 0$
Subtracting the last equation from first and multiplying the
last equation by (9/7) and subtracting from the second equation
we obtain $- 6 a + b = 0, 7 b + 6 c = 0$
$\Rightarrow \frac{a}{1} = \frac{b}{6} = \frac{c}{- 7} = k$ (say)
Thus, $a = k, b = 6 k, c = - 7 k$ .
As $(a, b, c) = (k, 6 k, - 7 k)$ lies on $2 x + y + z = 1$ , we get $k = 1$ .
Thus, $7 a + b + c = 7 k + 6 k - 7 k = 6 k = 6$

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