Assignment 1 โ Answers โ Q5
5. Consider two equations describing the relationship between three variables X, Y1, and Y2:
X = a + b Y1, (1)
X = c โ d Y2, (2)
where a, b, c, and d, are positive, known constants. The objective is to find values of X, Y1, and Y2 for
which both equations are satisfied and as well Y1 = Y2.
a. How many unknowns are there? Clearly identify them. How many equations are there?
Clearly identify them. Note that there are as many equations as unknowns. (If you do not see
this, and you should think more carefully). Thus, a unique solution exists. (Try to convince
yourself that having only (1) and (2) makes it impossible to solve for the unknown variables,
by equating X from (1) to X from (2) and trying to solve for Y1 and Y2.)
There are three unknowns: Y1, Y2, and X
There are three equations:
X = a + b Y1, (1)
X = c โ d Y2, (2)
Y1 = Y2. (3)
b. Replace both sides of the third equation (which one is it?) using expressions (1) and (2) to
solve for X.
๐
1= ๐
2โ1
๐(๐ โ ๐)=1
๐(๐ โ ๐)
โ ๐(๐ โ ๐)= ๐(๐ โ ๐)
โ๐๐ โ๐๐ =๐๐ โ๐๐
โ(๐ + ๐)๐ = ๐๐ +๐๐
โ ๐ = ๐๐ +๐๐
๐ + ๐
c. After solving for X, find the values of Y1 and Y2 and verify whether the third equation (again,
which one is it?) is satisfied.
Using ๐ = ๐๐+๐๐
๐+๐ and equation (1):
๐ = ๐๐ +๐๐
๐ + ๐ โ ๐
1=๐ โ ๐
๐ + ๐
Using ๐ = ๐๐+๐๐
๐+๐ and equation (1):
๐ = ๐๐ +๐๐
๐ + ๐ โ ๐
2=๐ โ ๐
๐ + ๐
