if you could exactly afford either 4 units of x and 27 units of y, or 9 units of x and 7 units of y, then if you spent all of your income on y, how many units of y could you buy?
A)23
B)63
C)43
D)13
E)None of the above.

Question

Quizplus · Accepted Answer

Set up two equations with x and y as variables:
4x + 27y = income and
9x + 7y = income
Solve for y in both equations:
y = (income - 4x) / 27 and
y = (income - 9x) / 7
Since we are spending all of our income on y, we can set these two equations equal to each other:
(income - 4x) / 27 = (income - 9x) / 7
Solving for income, we get:
income = 135x
Substituting this back into one of the original equations, we can solve for y:
4x + 27y = 135x
y = (131/27) x
Now we want to find out how many units of y we can buy with all of our income, so we need to find the value of x that maximizes y. We can do this by setting the derivative of y with respect to x equal to zero:
dy/dx = (131/27) = 0
Solving for x, we get:
x = 27
Now we can substitute this back into either equation to find the maximum value of y:
4(27) + 27y = income
y = 43
Therefore, we can buy 43 units of y with all of our income.

If You Could Exactly Afford Either 4 Units of X