Maths Olympiad - Algebra 3

Question:




In the above table each letter of the alphabet is given a value. The algebraic expression 4x − 3 is used as a key to convert the letters P S R X B O E into the word H A R M O N Y. Find the algebraic expression used to convert S R X B into G O L D ?



Answer:


For S R X B to become G O L D ,

S -> G where 4 -> 14
R -> O where 1 -> 5
X -> L where 3 -> 11
B -> D where 2 -> 8


Using the graphical y = mx + c expression, we will take the above numbers as x, y coordinates.

So, we have the following coordinates - (4,14), (1,5), (3,11) and (2,8).


Therefore,
y = mx + c
14 = 4m + c ----- sub. (4,14)
c = 14 - 4m ----- Eq. 1


y = mx + c
8 = 2m + c ---- sub. (2,8)
c = 8 - 2m ---- Eq. 2


Combine Eq.1 and Eq.2,
14 - 4m = 8 - 2m
6 = 2m
m = 3


Sub. m = 3 into Eq.2,
c = 8 - 2(3)
c = 8 - 6
c = 2


Sub. m = 3 and c = 2 into y = mx + c:
y = 3x + 2


Therefore, the algebraic expression used to convert S R X B into G O L D is:
3x + 2