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Solving Simultaneous Linear Equation Using Method of Elimination

If l (my + nz...

Question

If $l (m y + n z - l x) = m (n z + l x - m y) = n (l x + m y - n z)$ , prove $\frac{(y + z - x)}{l} = \frac{(z + x - y)}{m} = \frac{(x + y - z)}{n}$ .

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Solution

Step 1: Simplify the given equation
The given equation is,
$l (m y + n z - l x) = m (n z + l x - m y) = n (l x + m y - n z)$
On dividing the above equation by $l m n$ , we get,
$\frac{l (m y + n z - l x)}{l m n} = \frac{m (n z + l x - m y)}{l m n} = \frac{n (l x + m y - n z)}{l m n}$
$\Rightarrow \frac{(m y + n z - l x)}{m n} = \frac{(n z + l x - m y)}{l n} = \frac{(l x + m y - n z)}{l m}$
Step 2: Use the property of equal fractions
Now, since, $\frac{a}{b} = \frac{c}{d} = \frac{e}{f} \Rightarrow \frac{a + c}{b + d} = \frac{c + e}{d + f} = \frac{e + a}{f + b}$
So, $\frac{(m y + n z - l x) + (n z + l x - m y)}{m n + l n} = \frac{(n z + l x - m y) + (l x + m y - n z)}{l n + l m} = \frac{(l x + m y - n z) + (m y + n z - l x)}{l m + m n}$
$\Rightarrow$ $\frac{m y + n z - l x + n z + l x - m y}{m n + l n} = \frac{n z + l x - m y + l x + m y - n z}{l n + l m} = \frac{l x + m y - n z + m y + n z - l x}{l m + m n}$
$\Rightarrow \frac{2 n z}{(m + l) n} = \frac{2 l x}{(n + m) l} = \frac{2 m y}{(l + n) m}$
$\Rightarrow \frac{2 z}{(m + l)} = \frac{2 x}{(n + m)} = \frac{2 y}{(l + n)}$
Step 3: Use properties of equations
Now, on adding and subtracting the fractions in the above equation, we get,
$\frac{(2 z) + (2 y) - (2 x)}{(m + l) + (l + n) - (n + m)} = \frac{(2 z) + (2 x) - (2 y)}{(m + l) + (n + m) - (l + n)} = \frac{(2 x) + (2 y) - (2 z)}{(l + n) + (n + m) - (l + m)}$
$\Rightarrow \frac{2 z + 2 y - 2 x}{m + l + l + n - n - m} = \frac{2 z + 2 x - 2 y}{m + l + n + m - l - n} = \frac{2 x + 2 y - 2 z}{l + n + n + m - l - m}$
$\Rightarrow \frac{2 (z + y - x)}{2 l} = \frac{2 (z + x - y)}{2 m} = \frac{2 (x + y - z)}{2 n}$
$\Rightarrow \frac{(z + y - x)}{l} = \frac{(z + x - y)}{m} = \frac{(x + y - z)}{n}$
Hence, it is proved that $\frac{(y + z - x)}{l} = \frac{(z + x - y)}{m} = \frac{(x + y - z)}{n}$ .

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If lmy+nz−lx=mnz+lx−my=nlx+my−nz, prove y+z−xl=z+x−ym=x+y−zn.

If $l (m y + n z - l x) = m (n z + l x - m y) = n (l x + m y - n z)$ , prove $\frac{(y + z - x)}{l} = \frac{(z + x - y)}{m} = \frac{(x + y - z)}{n}$ .