Introduction
Last month, Maria embarked on an exciting hiking adventure, covering a total of 90 miles on two trails: a 5-mile mountain trail and a canal trail. As we delve into the world of mathematics, we'll explore the relationship between the number of times Maria hiked each trail and the total distance she covered. In this article, we'll use algebraic equations to represent the situation and solve for the unknown variables.
The Problem
Let $x$ represent the number of times Maria hiked the mountain trail, and let $y$ represent the number of times she hiked the canal trail. We know that the total distance covered by Maria is 90 miles, and each hike on the mountain trail covers 5 miles, while each hike on the canal trail covers an unknown distance, which we'll call $d$ miles. We can represent the situation using the following system of equations:
5x+dy=90
x+y=n
where $n$ is the total number of hikes Maria made.
Understanding the Equations
The first equation, $5x + dy = 90$, represents the total distance covered by Maria. The term $5x$ represents the distance covered by hiking the mountain trail, while the term $dy$ represents the distance covered by hiking the canal trail. The second equation, $x + y = n$, represents the total number of hikes Maria made.
Solving the System of Equations
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution. We can solve the second equation for $y$ in terms of $x$:
y=n−x
Substituting this expression for $y$ into the first equation, we get:
5x+d(n−x)=90
Expanding and simplifying the equation, we get:
5x+dn−dx=90
Combine like terms:
5x+dn−dx=90
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5x + dn<br/>
**Maria's Hiking Adventure: A Mathematical Exploration - Q&A**
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**Introduction**
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In our previous article, we explored the relationship between the number of times Maria hiked the mountain trail and the canal trail, and the total distance she covered. We used algebraic equations to represent the situation and solve for the unknown variables. In this article, we'll answer some frequently asked questions related to Maria's hiking adventure.
**Q: What is the total distance Maria covered on her hiking adventure?**
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A: According to the problem, Maria covered a total of 90 miles on her hiking adventure.
**Q: How many times did Maria hike the mountain trail?**
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A: Let's call the number of times Maria hiked the mountain trail $x$. We can represent the distance covered by hiking the mountain trail as $5x$ miles.
**Q: How many times did Maria hike the canal trail?**
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A: Let's call the number of times Maria hiked the canal trail $y$. We can represent the distance covered by hiking the canal trail as $dy$ miles.
**Q: What is the relationship between the number of times Maria hiked the mountain trail and the canal trail?**
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A: According to the problem, the total number of hikes Maria made is represented by the equation $x + y = n$. This means that the number of times Maria hiked the mountain trail is equal to the number of times she hiked the canal trail plus the total number of hikes she made.
**Q: How can we solve for the unknown variables?**
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A: We can use the method of substitution or elimination to solve for the unknown variables. Let's use the method of substitution. We can solve the second equation for $y$ in terms of $x$:
$y = n - x
Substituting this expression for $y$ into the first equation, we get:
5x+d(n−x)=90
Expanding and simplifying the equation, we get:
5x+dn−dx=90
Combine like terms:
5x+dn−dx=90
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