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 pur... 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.
Understand the Problem
The question is asking to solve a system of equations based on the number of soda and juice bottles purchased by David. The equations are derived from the total count of bottles and the total sugar content.
Answer
David purchased $x = 6$ bottles of soda and $y = 8$ bottles of juice.
Answer for screen readers
David purchased $x = 6$ bottles of soda and $y = 8$ bottles of juice.
Steps to Solve
- Define the variables
Let:
- $x$ = number of bottles of soda
- $y$ = number of bottles of juice
- Set up the equations
From the problem, we can derive the following two equations:
- Total bottles equation:
$$ x + y = 14 $$ - Total sugar content equation:
$$ 50x + 25y = 500 $$
- Rearrange the first equation
Solve the first equation for $y$:
$$ y = 14 - x $$
- Substitute into the second equation
Replace $y$ in the second equation with $(14 - x)$:
$$ 50x + 25(14 - x) = 500 $$
- Simplify and solve for $x$
Distribute and simplify:
$$ 50x + 350 - 25x = 500 $$
Combine like terms:
$$ 25x + 350 = 500 $$
Subtract 350 from both sides:
$$ 25x = 150 $$
Divide by 25:
$$ x = 6 $$
- Find $y$
Substitute $x = 6$ back into the equation for $y$:
$$ y = 14 - 6 = 8 $$
- Conclusion
David purchased 6 bottles of soda and 8 bottles of juice.
David purchased $x = 6$ bottles of soda and $y = 8$ bottles of juice.
More Information
David’s total of 14 bottles was comprised of 6 soda bottles and 8 juice bottles, which together contained 500 grams of sugar. Each soda bottle containing 50 grams and each juice 25 grams were essential in forming the system of equations.
Tips
- Mixing up equations: Ensure you correctly identify which equation represents total counts versus total sugar.
- Forgetting to distribute: When substituting values, always remember to properly distribute and combine like terms.
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