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Algebra Level 3

If w − x y − z = 2 \dfrac{w-x}{y-z} = 2 and w − y x − z = 3 \dfrac{w-y}{x-z} = 3 , then determine the value of ∣ w − z x − y ∣ . \large \left|\frac{w-z}{x-y} \right| .


The answer is 5.

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Ahmad Khamis by Ahmad Khamis , 24, Saudi Arabia

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4 solutions

{ w − x y − z = 2 ⇒ w − x = 2 y − 2 z . . . ( 1 ) w − y x − z = 3 ⇒ w − y = 3 x − 3 z . . . ( 2 ) \begin{cases} \dfrac{w-x}{y-z} = 2 & \Rightarrow w - x = 2y - 2z &...(1) \\ \dfrac{w-y}{x-z} = 3 &\Rightarrow w-y = 3x - 3z &...(2) \end{cases}

{ ( 1 ) − ( 2 ) : − x + y = 2 y − 2 z − 3 x + 3 z ⇒ z = 2 x − y . . . ( 3 ) ( 3 ) → ( 1 ) : w − x = 2 y − 4 x + 2 y ⇒ w = 4 y − 3 x . . . ( 4 ) \begin{cases} (1) - (2): & - x + y = 2y - 2z - 3x + 3z & \Rightarrow z = 2x - y & ... (3) \\ (3) \rightarrow (1): & w - x = 2y - 4x + 2y & \Rightarrow w = 4y - 3x &...(4) \end{cases}

⇒ ∣ w − z x − y ∣ = ∣ 4 y − 3 x − 2 x + y x − y ∣ = ∣ 5 y − 5 x x − y ∣ = 5 \Rightarrow \left| \dfrac{w-z}{x-y} \right| = \left| \dfrac{4y - 3x-2x + y}{x-y} \right| = \left| \dfrac{5y - 5x}{x-y} \right| = \boxed{5}

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Khushal Sethi
Aug 7, 2015

Apply componendo and dividendo in both equations.divide them and again apply c&d. We get answer

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Moderator note:

Right. Can you elaborate on it?

Ankush Gogoi
Aug 6, 2015

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Mridul Gupta
Aug 6, 2015

Just assume and substitute value of y=4 and z=3 you will get first equation as w=2+x and second equation as w-4 = 3 (x-3) solve both equations for w=11/2 and x=7/2 and substitute all 4 value in the expression and get answer as 5.

This method works because looking at the equation we can see that w,x,y,z belong to R and if we assume value for two variables then we will get the value of the other two variables accordingly.

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How do we know if this would be true for all possible values that satisfy the initial conditions?

Calvin Lin Staff - 5 years, 10 months ago

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