Question

1. you are in charge of decorating the gym for the homecoming dance. you purchased 6 bags of balloons and 5 bags of large stars all for $22.15. you soon realized that this was not enough to decorate the whole gym. on your second trip to the store, you bought 8 bags of balloons and 2 bags of large stars all for $20.90. what was the price for each item?

define variables:
b balloons s stars
write equations:
6b + 5s = 22.15
8b + 2s = 20.90
check solution:
solve:
state solution in sentence form:
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Explanation:

Step1: Define variables and equations

Let \( B \) be the price of a bag of balloons (in dollars) and \( S \) be the price of a bag of large stars (in dollars). We have the system of equations:
\[

$$\begin{cases} 6B + 5S = 22.15 \\ 8B + 2S = 20.90 \end{cases}$$

\]

Step2: Eliminate one variable

Multiply the first equation by 2 and the second equation by 5 to make the coefficients of \( S \) equal:
First equation: \( 12B + 10S = 44.30 \)
Second equation: \( 40B + 10S = 104.50 \)
Subtract the first new equation from the second new equation:
\( (40B + 10S) - (12B + 10S) = 104.50 - 44.30 \)
\( 28B = 60.20 \)

Step3: Solve for \( B \)

Divide both sides by 28: \( B=\frac{60.20}{28} = 2.15 \)

Step4: Solve for \( S \)

Substitute \( B = 2.15 \) into the second original equation \( 8(2.15)+2S = 20.90 \)
\( 17.20+2S = 20.90 \)
Subtract 17.20 from both sides: \( 2S = 20.90 - 17.20 = 3.70 \)
Divide by 2: \( S=\frac{3.70}{2}=1.85 \)

Step5: Check the solution

Substitute \( B = 2.15 \) and \( S = 1.85 \) into the first equation:
\( 6(2.15)+5(1.85)=12.90 + 9.25 = 22.15 \) (matches the first total)
Substitute into the second equation:
\( 8(2.15)+2(1.85)=17.20+3.70 = 20.90 \) (matches the second total)

Answer:

The price of a bag of balloons is \$2.15 and the price of a bag of large stars is \$1.85.

State solution in sentence form:
The price for each bag of balloons is \$2.15, and the price for each bag of large stars is \$1.85.