Question

linear equations and inequalities
helena needs 3.5 cups of flour per loaf of bread and 2.5 cups of flour per batch of muffins. she also needs 0.75 cup of sugar per loaf of bread and 0.75 cup of sugar per batch of muffins. helena has 17 cups of flour and 4.5 cups of sugar available for baking
which combination of loaves of bread and batches of muffins could helena bake?
4 loaves of bread and 2 batches of muffins
2 loaves of bread and 4 batches of muffins
3 loaves of bread and 3 batches of muffins
5 loaves of bread and 1 batch of muffins

Explanation:

Step1: Define Variables

Let \( b \) = loaves of bread, \( m \) = batches of muffins. Flour constraint: \( 3.5b + 2.5m \leq 17 \). Sugar constraint: \( 0.75b + 0.75m \leq 4.5 \).

Step2: Check Option 1 (4,2)

Flour: \( 3.5(4) + 2.5(2) = 14 + 5 = 19 \). \( 19 > 17 \) (fails flour). Sugar: \( 0.75(4) + 0.75(2) = 3 + 1.5 = 4.5 \) (passes sugar). Reject.

Step3: Check Option 2 (2,4)

Flour: \( 3.5(2) + 2.5(4) = 7 + 10 = 17 \) (passes flour). Sugar: \( 0.75(2) + 0.75(4) = 1.5 + 3 = 4.5 \) (passes sugar). Both constraints satisfied.

Step4: Verify Other Options (Optional)

Option 3 (3,3): Flour: \( 3.5(3) + 2.5(3) = 10.5 + 7.5 = 18 \). \( 18 > 17 \) (fails flour). Option 4 (5,1): Flour: \( 3.5(5) + 2.5(1) = 17.5 + 2.5 = 20 \). \( 20 > 17 \) (fails flour).

Answer:

2 loaves of bread and 4 batches of muffins