Given the real numbers m and n such that m+4n=5 and 2m+4n=6 find what is 3m+4n.

The request is somehow intuitive. We can easily notice
that 3m + 4n = 7, since m+4n=5 and 2m+4n=6. This means that
m=n=1.

But, we'll prove that 3m + 4n = 7, solving the
system of equations m+4n=5 and 2m+4n=6.

We'll apply
elimination method. For this reason, we''ll multiply the 1st equation by -2 and we'll
add the resulting equation to the 2nd.

-2m - 8n + 2m + 4n =
-10 + 6

We'll eliminate m and we'll combine like
terms:

-4n = -4

n =
1

We'll substitute n = 1 into the 1st
equation:

m + 4 = 5

m = 5 -
4

m = 1

Now, we'll calculate
3m+4n:

3m+4n = 3*1 + 4*1 =
7

The result of the expression 3m+4n, for m
= n = 1, is 3m+4n = 7.