Question

x square is equal to y +z and y square is equal to z+x and z square is equal to x+y,find value of( 1 by x+1)+(1 by y+1)+(1 by z+1)

Answer 1

Question

x² = y + z and y² = z + x and z² = x + y. Find value of 1/(x + 1) + 1/(y + 1) + 1/(z + 1)

Answer

1/(x + 1) + 1/(y + 1) + 1/(z + 1) = 1 or 3

Explanation-

x² = y + z............(1)

y² = z + x............(2)

z² = x + y.............(3)

Let us assume that, x = y = z = 0

Put in eq (1)

(0)² = 0 + 0

0 = 0 + 0

Similarly, put in eq (2) and (3)

(0)² = 0 + 0

0 = 0

Now, substitute value of x, y and z = 0 in 1/(x + 1) + 1/(y + 1) + 1/(z + 1)

→ 1/(0 + 1) + 1/(0 + 1) + 1/(0 + 1)

→ 1/1 + 1/1 + 1/1

→ 3/1 = 3

Similarly, assume that x = y = z = 2

Substitute value of x, y and z in eq (1), (2) and (3).

By doing this we get,

(2)² = 2 + 2

4 = 4

Now, substitute value of x, y and z = 2 in 1/(x + 1) + 1/(y + 1) + 1/(z + 1)

→ 1/(2 + 1) + 1/(2 + 1) + 1/(2 + 1)

→ 1/3 + 1/3 + 1/3

→ 3/3 = 1

Answer 2

Given :-

• x² = (y + z)
• y² = (z + x)
• z² = (x + y)

To Find :-

• 1/(x + 1) + 1/(y + 1) + 1/(z + 1) = ?

Solution :-

1/(x + 1) + 1/(y + 1) + 1/(z + 1)

Multiply both the numerator and denominator of the first term by x, second term by y, third term by z , we get :-

→ (x/x)(1/x+1) + (y/y)(1/y+1) + (z/z)(1/z+1)

→ x /(x²+x) + y/(y²+y) + z/(z² + z)

Now, Putting value of x², y² & z² From Given we get,

→ x/(y+z+x) + y/(z+x+y) + z/(x+y+z)

Taking LCM now, we get,

→ (x + y + z) / (x + y + z)

→ 1 (Ans).

Hence, Value of 1/(x + 1) + 1/(y + 1) + 1/(z + 1) will be 1.

(This is The simplest Method).

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Shortcut :-

Assume That, x = y = z = 2.

Check Given values :-

→ x² = (y + z) => 2² = (2 + 2) => 4 = 4 = Satisfy

→ y² = (z + x) => 2² = (2 + 2) => 4 = 4 = Satisfy

→ z² = (x + y) => 2² = (2 + 2) => 4 = 4 = Satisfy

Hence , Value of :-

→ 1/(x + 1) + 1/(y + 1) + 1/(z + 1)

→ 1/(2+1) + 1/(2+1) + 1/(2+1)

→ 1/3 + 1/3 + 1/3

→ 3/3

→ 1 (Ans).

Hence, Value of 1/(x + 1) + 1/(y + 1) + 1/(z + 1) will be 1.

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