Final answer:
The problem of finding the price of 1 cup of dried fruit and 1 cup of almonds results in a system of linear equations that are multiples of each other, indicating that they represent the same line. Therefore, there is no unique solution, as the equations suggest the prices of dried fruit and almonds are proportional, and many combinations of prices will satisfy the given conditions.
Step-by-step explanation:
To find the price of 1 cup of dried fruit and 1 cup of almonds using a system of linear equations, we need to set up two equations based on the given information. Let's denote x as the price per cup of dried fruit and y as the price per cup of almonds.
The first equation comes from the small bag: 3x + 4y = 6. This signifies that 3 cups of dried fruit plus 4 cups of almonds equals the total cost of the small bag, $6.
The second equation is formed with the large bag: 4.5x + 6y = 9. This means 4.5 cups of dried fruit plus 6 cups of almonds equals the total cost of the large bag, $9.
Now, we have the following system of linear equations:
- 3x + 4y = 6
- 4.5x + 6y = 9
To solve this system, we can use either substitution or elimination. We'll use elimination in this case.
- Multiply the first equation by 1.5 to align the coefficient of y in both equations: 1.5(3x + 4y) = 1.5(6) becomes 4.5x + 6y = 9.
- Subtract the new equation from the second one: (4.5x + 6y) - (4.5x + 6y) = 9 - 9, which cancels out both variables, indicating that the equations are multiples of each other and represent the same line.
Since the equations represent the same line, we have an infinite number of solutions, meaning that any point on the line is a solution, which implies the prices of fruits and nuts are proportional to each other and can be many combinations as long as they satisfy the line equation.
Due to the equations being the same, we cannot find a unique solution for the prices of dried fruit and almonds.