## Equations for F

Given that
$A + B = C$
$C + D = E$
$E + A = F$
$B + D + F = 100$
$A = 8$
What is the value of F?
Source: mathcontest.olemiss.edu 3/3/2008

SOLUTION
Let’s label these five equations for ease in reference as follows:
$A+B=C$        (1)
$C+D=E$        (2)
$E+A=F$        (3)
$B+D+F=100$        (4)
$A=8$             (5)

Rewrite Eq. (3) and add it to Eq. (4):
$E+A-F=0$
$B+D+F=100$
————————-
$E+A+B+D=100$        (6)

Rewrite Eq. (1) and Eq. (2) and add them:
$A+B-C=0$
$C+D-E=0$
———————
$A+B+D-E=0$        (7)

Multiply Eq. (7) by $-1$ and add it to Eq. (6):
$E+A+B+D=100$
$-A-B-D+E=0$
—————————–
$2E=100$
$E=100\div 2=50$

Value of F
Substitute the values of $A,E$ into Eq. (3):
$E+A=F$
$50+8=F$
$58=F$