Question

unit 4.4 to 4.5 formative
5.) a school club is planning a fundraiser event. they need to buy two types of decorations: balloons and streamers. here are the constraints:

• balloons cost $2 each, and streamers cost $3 each.
• they have a maximum budget of $60 for decorations.
• they want to have at least 10 items in total.

a.) identify what the variables represent: (1 points)

x = _____________ y = _____________

b.) write a system of linear inequalities that represents this situation: (2 points)

inequality 1: _____________________________________

inequality 2: _____________________________________

c.) show whether the point (10, 10) is a solution to the given system of inequalities. explain your reasoning and steps. (3 points)

Explanation:

Step1: Define variables

Let $x$ = number of balloons, $y$ = number of streamers

Step2: Write budget inequality

$2x + 3y \leq 60$

Step3: Write total items inequality

$x + y \geq 10$

Step4: Test point (10,10) in inequality 1

Substitute $x=10, y=10$: $2(10) + 3(10) = 20 + 30 = 50$

Step5: Compare to budget limit

$50 \leq 60$, so this holds true.

Step6: Test point (10,10) in inequality 2

Substitute $x=10, y=10$: $10 + 10 = 20$

Step7: Compare to minimum items

$20 \geq 10$, so this holds true.

Answer:

a.)
$x$ = number of balloons
$y$ = number of streamers

b.)
Inequality 1: $2x + 3y \leq 60$
Inequality 2: $x + y \geq 10$

c.)
The point (10, 10) is a solution. When substituted into both inequalities, $2(10)+3(10)=50\leq60$ and $10+10=20\geq10$, so both constraints are satisfied.