Given the function f(x) = x+1, determine x and y if: f(x) + f(y)=7 f(x) + 2f(y)=12
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We have the function f(x) = x + 1
Now f(x) + f(y) = 7
=> x + 1 + y + 1 = 7
=> x + y = 5 ...(1)
f(x) + 2*f(y) = 12
=> x + 1 + 2y + 2 = 12
=> x + 2y = 9 ...(2)
(2) - (1)
=> y = 4
substitute y = 4 in (1)
=> x + 4 = 5
=> x = 1
Therefore x = 1 and y = 4.
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We'll note the equations of the system:
f(x)+f(y) = 7 (1)
f(x)+2f(y) = 12 (2)
We'll subtract (2) - (1):
f(x)+2f(y)-f(x)-f(y) = 12 - 7
We'll combine and eliminate like terms:
f(y) = 5
We'll substitute f(y) in (1):
f(x)+5 = 7
We'll subtract 5:
f(x) = 7 - 5
f(x) = 2
Now, we'll re-write f(x) = 2:
x + 1 = 2
We'll subtract 1:
x = 1
f(y) = 5
y + 1 = 5
y = 4
The solution of the given system is (1 , 4).
f(x)+f(y) = 7...........(1)
f(x)+2f(y) = 12.........(2).
(2) - (1): 2f(y)-f(y) = f(y) = 5.
2(1) - (2) = 2f(x)- f(x) = 2*7-12 = 14-12 = 2.
So f(x) = 2.
Also give f(x) = x+1. So x+1 = 2. Or x = 2-1 = 1.
So x = 1.
Since f(x) = x+1, f(y) = y+1. So y+1 = 5. Therefore y = 5-1 = 4.
Therefore x= 1 and y = 4.
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