# 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.**

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.