UKMT Special (Problem $16$)

Find all real numbers $x, y, z$ which satisfy the simultaneous equations

$x^2 - 4y + 7 = 0$

$y^2 - 6z + 14 = 0$

$z^2 - 2x - 7 = 0$

[UKMT BMO Round $1$ $2012$ , Q $3$ ]

#Algebra

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
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Comments

You have until next Friday, $3:00$ pm!

Yajat Shamji - 6 months, 4 weeks ago

Step 1) Add all of them and factor the term.

   x^2 - 2x + y^{2} - 4y + z^{2} - 6z + 14 = 0

   x^2 - 2x + 1 + y^2 - 4y + 4 + z^2 - 6z + 9 = 0

   (x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 0

Step 2) If any of the squared term is not 0, it must be a positive real number.

If there is at least one squared term that is not 0, the equation will be false because the right hand sight must be greater than 0.

So, each the square term must be equal to 0.

   (x - 1)^2 = 0    =>    x - 1  = 0    =>    x=1

   (y - 2)^2 = 0    =>    y - 2 = 0    =>    y=2

   (z - 3)^2 = 0    =>    z - 3 = 0    =>    z=3

(x , y , z)=(1 , 2 , 3) is the only real solution.

I apologize for the bad representation.

Quest Keeper - 6 months, 3 weeks ago

Correct method and solution!

Yajat Shamji - 6 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

UKMT Special (Problem \(16\))

Comments