Suppose you bought supplies for a party. Three rolls of steamers and 15 balloons cost $30. Later, you brought 2 rolls of steamers and 4 balloons for $11. How much did each roll of steamers cost? How much did each balloon cost?
To solve this problem, we can set up a system of equations based on the given information.
Let's assume the cost of each roll of streamers is "x" dollars, and the cost of each balloon is "y" dollars.
Based on the first purchase, we have the equation:
3x + 15y = 30
And based on the second purchase, we have the equation:
2x + 4y = 11
To determine the cost of each roll of streamers (x) and each balloon (y), we can solve this system of equations.
Method 1: Substitution Method
In this method, we solve one equation for one variable and substitute that expression into the second equation.
From the first equation, we can express x in terms of y:
3x = 30 - 15y
x = (30 - 15y)/3
x = 10 - 5y
Now we substitute the expression for x into the second equation:
2(10 - 5y) + 4y = 11
20 - 10y + 4y = 11
-6y = -9
y = (-9)/(-6)
y = 3/2 = 1.5
Now that we know the cost of each balloon is $1.5, we can substitute this value back into any of the original equations and solve for x.
Using the first equation:
3x + 15(1.5) = 30
3x + 22.5 = 30
3x = 30 - 22.5
3x = 7.5
x = 7.5/3
x = 2.5
Therefore, each roll of streamers costs $2.5 and each balloon costs $1.5.
Method 2: Elimination Method
In this method, we manipulate the equations to eliminate one of the variables.
We can multiply the second equation by 3, and the first equation by 2, to eliminate x:
6x + 12y = 33
6x + 30y = 60
Now, subtract the second equation from the first:
(6x + 12y) - (6x + 30y) = 33 - 60
-18y = -27
y = (-27)/(-18)
y = 1.5
Substituting the value of y into the first equation:
2x + 15(1.5) = 30
2x + 22.5 = 30
2x = 30 - 22.5
2x = 7.5
x = 7.5/2
x = 3.75
Therefore, each roll of streamers costs $3.75, and each balloon costs $1.5.