Question

david went to the grocery store and bought bottles of soda and bottles of juice. each bottle of soda has 50 grams of sugar and each bottle of juice has 25 grams of sugar. david purchased a total of 14 bottles of juice and soda which collectively contain 500 grams of sugar. graphically solve a system of equations in order to determine the number of bottles of soda purchased, x, and the number of bottles of juice purchased, y.

Explanation:

Step1: Define Variables and Equations

Let \( x \) be the number of soda bottles, \( y \) be the number of juice bottles.
From total bottles: \( x + y = 14 \) (Equation 1).
From total sugar: \( 50x + 25y = 500 \) (Equation 2).

Step2: Rewrite Equations in Slope-Intercept Form

For Equation 1: \( y = -x + 14 \).
For Equation 2: Divide by 25: \( 2x + y = 20 \) → \( y = -2x + 20 \).

Step3: Graph the Lines

• Line 1 (\( y = -x + 14 \)): Intercepts at \( (0,14) \) and \( (14,0) \).
• Line 2 (\( y = -2x + 20 \)): Intercepts at \( (0,20) \) and \( (10,0) \).

Step4: Find Intersection

The lines intersect where \( -x + 14 = -2x + 20 \).
Solve: \( x = 6 \), then \( y = -6 + 14 = 8 \).

Answer:

The number of soda bottles \( x = 6 \), and juice bottles \( y = 8 \).