You buy only apples and bananas. Your budget is such that you can purchase 3 apples and 4 bananas or 9 apples and 2 bananas. Write down the equation for the budget line with bananas on the y-axis. Is that equation unique?
I believe the answer is:
2y+9x=4y+3x yes, it is unique because the two set points will not change regardless of income
Hummm.
I think the equation is y=5-(1/3)x. Its unique because no other linear equation will hit both known possible consumption points (4,3) and (9,2)
To write down the equation for the budget line, we need to assign variables to the quantities of apples and bananas. Let's use x for the number of apples and y for the number of bananas.
According to the given information, if you purchase 3 apples and 4 bananas, or if you purchase 9 apples and 2 bananas, both options will fit within your budget.
Let's calculate the slope of the budget line by using the formula:
slope = (change in y) / (change in x) = (y₂ - y₁) / (x₂ - x₁)
Using the first set of points (3 apples, 4 bananas), and the second set of points (9 apples, 2 bananas), we can calculate the slope:
slope = (2 - 4) / (9 - 3) = -2 / 6 = -1/3
Now, we can use the point-slope form of the equation of a line to find the equation of the budget line:
y - y₁ = m(x - x₁)
Using one of the given points, let's say (3, 4), and the slope we calculated:
y - 4 = (-1/3)(x - 3)
Simplifying this equation gives:
3y + 9x = 4y + 12
Subtracting 3y from both sides:
9x = 4y + 12 - 3y
Combining like terms:
9x = y + 12
Rearranging the equation to have the bananas on the y-axis gives:
y = 9x - 12
So, the equation for the budget line with bananas on the y-axis is y = 9x - 12.
To answer the second part of your question, the equation for the budget line is indeed unique. It represents all the possible combinations of apples and bananas that can be purchased with the given budget constraint, and it will not change regardless of your income.