Question
Question: If \(x + y = 9\) , \(y + z = 7\) and \(z + x = 5\) then A) \(x + y + z = 10\) B) Arithmetic mea...
If x+y=9 , y+z=7 and z+x=5 then
A) x+y+z=10
B) Arithmetic mean of x, y, z is 3.5
C) Median of x, y, z is 3.5
D) x+y+z=10.5
Solution
We can take any two equations and write them in terms of the common variable. Then we can substitute them in the third equation and find the value of one variable. Then we can find the value of other variables by substituting back in the equations. Then we can find their sum. Then we can divide the sum with 3 to find the mean. To find the median, we can arrange them in ascending order and the middle number will give the median. Then we can compare with the options.
Complete step by step solution:
We have the equations
x+y=9 ,
y+z=7 and
z+x=5
Consider the equation x+y=9 .
On rearranging, we get,
⇒x=9−y … (1)
Now consider the equation y+z=7 .
On rearranging, we get,
⇒z=7−y … (2)
Now take the 3rd equation z+x=5
On substituting equations (1) and (2), we get,
⇒7−y+9−y=5
On simplification, we get,
⇒16−2y=5
On rearranging, we get,
⇒2y=16−5
On simplification we get,
⇒2y=11
On dividing throughout with 2, we get,
⇒y=211
So, we have,
⇒y=5.5 … (3)
On substituting in equation (1), we get,
⇒x=9−5.5
On simplification, we get,
⇒x=3.5
On substituting equation (3) in equation (2), we get,
⇒z=7−5.5
On simplification, we get,
⇒z=1.5
Now we can find the sum of the 3 numbers.
⇒x+y+z=3.5+5.5+1.5
On simplification, we get,
⇒x+y+z=10.5 … (4)
Now we can find the arithmetic mean.
⇒mean=3x+y+z
On substituting equation (4), we get,
⇒mean=310.5
On simplification, we get,
⇒mean=3.5 … (5)
Now we can check the median.
For that we can arrange the values in the ascending order.
⇒ 1.5,3.5,5.5
So, the middle most observation is 3.5.
So, the median is 3.5… (6)
From (4), (5) and (6), the correct options are,
Option B, Arithmetic mean of x, y, z is 3.5
Option C, Median of x, y, z is 3.5
Option D, x+y+z=10.5
Note:
Note: Alternate method to find the sum and mean is given by,
We have the equations
x+y=9 ,
y+z=7 and
z+x=5
On adding all the equations, we get,
⇒x+y+y+z+z+x=9+7+5
On simplification, we get,
⇒2x+2y+2z=21
On dividing throughout with 2, we get,
⇒x+y+z=221
⇒x+y+z=10.5 … (a)
Now we can find the arithmetic mean.
⇒mean=3x+y+z
On substituting equation (a), we get,
⇒mean=310.5
On simplification, we get,
⇒mean=3.5
For finding the median, we must find the values of the variables. So, we cannot find it using this shortcut method.