How many bottles of each type of juice did you buy if you spent $21.50?

Question

1.        You bought  bottles of grape juice and lemonade.  The grape juice was on sale for  per bottle, and the lemonade was on sale for  per bottle. How many bottles of each type of juice did you buy if you spent ?

Answer 1

X = total bottles purchased at total $21.50

Y = bottles of grape juice at A per bottle

Z = bottles of lemonade at B per bottle

That gives two equations:

21.5X = AY + BZ

X=Y+Z

Give some more information, such as values for A and B, reducing the system of equations to two equations with two unknowns and you can solve it, by matrix algebra, simple algebra or just substitution

Y=X-Z, so 21.5X=A(X-Z) + BZ substitute X-Z for Y

21.5X-AX = AZ+BZ

X = (B-A))Z/(21.5-A)

Z=(21.5-A)X/(B-A) If you know A,B and X, you can solve for Z= the number of lemonade bottles,

then when you know X and Z you know Y=X-Z= number of grape juice bottles

If you have just 2 equations, and 3 or more unknowns, there usually is no unique solution.