Trail Maps and Elevation Systems

Use two linear equations to plan and check a hiking trip.

Plan the Route

Write a system for each situation. Solve using substitution or elimination, and include units in your work.

A 9 km hiking route has only uphill and downhill sections. Each uphill kilometer gains 200 m of elevation, and each downhill kilometer loses 100 m. The finish is 600 m higher than the start. How many kilometers of each type does the route contain?

Uphill: km; downhill: km

Nadia plans a 14 km hike with uphill and downhill sections. She walks uphill at 2 km per hour and downhill at 4 km per hour. Her walking time, not including breaks, is 5 hours. How far does she walk uphill and downhill?

Uphill: km; downhill: km

A hiking guide builds two routes from repeated sections of Pine Trail and Ridge Trail. Every full Pine section has the same length, and every full Ridge section has the same length. Use the route totals to find each section's length.

Route Distances
Pine sections	Ridge sections	Total distance (km)
2	1	17
1	2	19

One Pine section: km; one Ridge section: km

Two trails leave the same vertical map line. On the map, x is horizontal distance east in kilometers and h is elevation in meters. Summit Trail starts at 350 m and gains 80 m per kilometer east. Valley Trail starts at 710 m and loses 40 m per kilometer east. Which system can locate where the trails have the same map position and elevation?

A. h = 80 + 350x and h = 40 − 710xB. h = 350 + 80x and h = 710 − 40xC. h = 350 − 80x and h = 710 + 40xD. h = 350 + 80x and h = 710 + 40x

Locate Hikers and Junctions

Solve both equations together. Check that your answer makes sense for the map or hiking situation.

On a trail map, x and y measure kilometers east and north of a ranger station. Two straight trail sections follow y = x + 1 and y = −x + 5. Both sections extend through their intersection. Find the junction's coordinates.

x = ; y =

Luis and Amira start at opposite ends of a 21 km trail at the same time and walk toward each other without stopping. Luis walks at 3 km per hour, and Amira walks at 4 km per hour. Let t be hours after they start and d be distance from Luis's starting point. Their positions satisfy d = 3t and d = 21 − 4t. When and where do they meet?

They meet after hours, km from Luis's starting point.

During a planned hike, one group climbs from an elevation of 420 m at 60 m per hour. Another group descends from 780 m at 30 m per hour. Both start at the same time and keep these rates for 5 hours. At what time do they have the same elevation? What is that elevation? The groups are on different trails, so equal elevations do not necessarily mean they meet.

Time after starting: hours; elevation: m

A route has two sections. Let x and y be their signed elevation changes in meters: positive means a gain, and negative means a loss. The total change is −40 m. The first section's signed change is 160 m greater than the second section's signed change. Thus, x + y = −40 and x − y = 160. Find both signed changes.

First section: m; second section: m

Two groups' elevations t hours after starting are modeled by h = 500 + 40t and h = 560 + 40t. What does this system tell you?

A. The groups have equal elevations at every time.B. The groups have equal elevations after 1.5 hours.C. The groups never have equal elevations in these models.D. The groups have equal elevations after 60 hours.

Check the Map and the Model

Use systems to check map claims and hiking plans. Explain your reasoning in the final question.

10.

Two map files describe straight trails with y = 2x − 3 and 2y = 4x − 6. Both descriptions cover 0 ≤ x ≤ 4. How are the mapped trails related?

A. They overlap at every point throughout the interval.B. They remain parallel and separate throughout the interval.C. They cross at exactly one point in the stated interval.D. They touch only at the two ends of the interval.

11.

A waypoint lies at the intersection of two straight map lines. Its coordinates satisfy x − y = −4 and 3x + 2y = 18. Coordinates are measured in kilometers from the map origin. Find the waypoint.

x = ; y =

12.

Jules leaves a trailhead and walks at 3 km per hour. One hour later, Ren leaves the same trailhead on the same route at 4.5 km per hour. Neither stops, and the trail continues for 15 km. Let t be hours after Ren starts and d be distance from the trailhead. Use d = 3(t + 1) and d = 4.5t to find when Ren catches Jules and how far they have traveled.

Ren catches Jules hours after Ren starts, km from the trailhead.

13.

A map shows a short trail following y = x + 2 only for 0 ≤ x ≤ 2. A second trail follows y = 2x − 1 only for 0 ≤ x ≤ 4. Mateo solves the two equations and says the actual trails meet at (3, 5). Is his claim correct? Use both the equations and the stated trail limits to explain.