Solving a Cycling Problem Using Systems of Equations

In this article, we will solve a cycling problem using the method of systems of equations. This problem involves determining the time spent at different speeds by a cyclist based on the total distance and average speed. Here's an in-depth look at the solution:

Problem Statement

A cyclist travels for x hours at 5 km/h and y hours at 10 km/h. The cyclist travels 35 km altogether and has an average speed of 7 km/h. We need to find the values of x and y.

Solution Approach

The problem can be solved using a system of equations. We will set up and solve this system through a step-by-step process.

Distance Equation

The total distance traveled by the cyclist is given by the sum of the distances traveled at each speed. The distance can be calculated using the formula:

Distance Speed x Time

Therefore, the total distance equation is:

5x 10y 35

Average Speed Equation

The average speed is calculated as the total distance divided by the total time. The total time can be expressed as x y hours. Hence, the average speed equation is:

Average Speed Total Distance / Total Time

Substituting the given values, we get:

35}{x y} 7

Multiplying both sides by x y, we can rearrange this to:

35 7(x y)

Simplifying this, we get:

x y 5

Step-by-Step Solution

Step 1: Solve for One Variable

From the second equation, we can express y in terms of x:

y 5 - x

Step 2: Substitute into the First Equation

Substituting y into the first equation:

5x 10(5 - x) 35

Expanding this gives:

5x 50 - 1 35

Combining like terms results in:

-5x 50 35

Subtracting 50 from both sides:

-5x -15

Dividing by -5:

x 3

Step 3: Find y

Now substitute x back into the equation for y:

y 5 - x 5 - 3 2

Conclusion

The values of x and y are:

boxed{x 3 , hours , y 2 , hours}

Alternative Approaches

Another method to solve this problem involves using the concept of ratios and the relationship between average speed, distances, and travel times. If the average speed is 7 km/hr and the speeds are 5 km/hr and 10 km/hr, the ratio y:x is given by:

7 - 5 : 10 - 7 2 : 3

TOTAL TIME x y 35 km / 7 km/hr 5 hours. Splitting 5 hours in the ratio 2:3, we get:

y 2 hours and x 3 hours

This confirms that x 3 hours and y 2 hours.

Key Concepts

1. **Systems of Equations:

A set of two or more equations with a same set of unknowns.

2. **Average Speed:

Total distance divided by total time.

3. **Ratio:

The relative size of two quantities expressed as the quotient of one divided by the other.

By combining the principles of systems of equations and ratio analysis, we can solve complex problems involving travel times and speeds.