A store is having a sale on jelly beans and trail mix. For 3 pounds of jelly beans and 5 pounds of trail mix, the total cost is $17. For 9 pounds of jelly beans and 7 pounds of trail mix, the total cost is $37. Find the cost for each pound of jelly beans and each pound of trail mix.
Let's start by defining some variables:
- Let j be the cost per pound of jelly beans
- Let t be the cost per pound of trail mix
We can use these variables to write two equations based on the information given in the problem:
3j + 5t = 17 (for 3 pounds of jelly beans and 5 pounds of trail mix)
9j + 7t = 37 (for 9 pounds of jelly beans and 7 pounds of trail mix)
We now have a system of two equations with two variables. We can solve for j and t using any method of our choice (substitution, elimination, etc.). Here, we'll use elimination by multiplying the first equation by 3 and the second equation by -1, then adding them together:
9j + 15t = 51
-9j - 7t = -37
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8t = 14
Dividing both sides by 8, we get t = 1.75. Now we can substitute this value back into one of the original equations (let's use the first one) to solve for j:
3j + 5(1.75) = 17
3j + 8.75 = 17
3j = 8.25
j = 2.75
Therefore, the cost per pound of jelly beans is $2.75 and the cost per pound of trail mix is $1.75.
To find the cost per pound of jelly beans and trail mix, let's set up a system of equations using the given information.
Let's assume the cost per pound of jelly beans is 'x' and the cost per pound of trail mix is 'y'.
From the first equation, we know that:
3x + 5y = 17 (1)
From the second equation, we know that:
9x + 7y = 37 (2)
We now have a system of two equations. We can solve this system by using the method of substitution or elimination.
Let's use the method of substitution:
First, solve equation (1) for x in terms of y:
3x = 17 - 5y
x = (17 - 5y) / 3
Now, substitute this value of x into equation (2):
9((17 - 5y) / 3) + 7y = 37
Simplify the equation:
(51 - 15y) + 21y = 37
51 - 15y + 21y = 37
6y = -14
y = -14 / 6
y = -7/3
Now, substitute the value of y back into equation (1) to find the value of x:
3x + 5(-7/3) = 17
3x - 35/3 = 17
3x = 17 + 35/3
3x = 51/3 + 35/3
3x = 86/3
x = (86/3) / 3
x = 86/9
x = 86/9
Therefore, the cost per pound of jelly beans is $86/9 or $9.56 (rounded to two decimal places), and the cost per pound of trail mix is -$7/3 or -$2.33 (rounded to two decimal places).