A group of hikers is progressing steadily along an uphill trail. The graph shows their elevation (or height above sea level), in feet, at each distance from the start of the trail, in miles.

Graph of a line, origin O. Horizontal axis, distance along trail in miles, scale is 0 to 9, by 1’s. Vertical axis, elevation in feet, scale is 0 to three thousand, by 500’s. Line starts at 500 comma 0, passes through 2 comma one thousand, 4 comma one thousand five hundred, 6 comma two thousand and 8 comma two thousand five hundred.

1. What is the slope of the graph? Show your reasoning.
2. What does the slope tell us about this situation?
3. Write an equation that represents the relationship between the hikers' distance from the start of the trail, , and their elevation, .
4. Does the equation   represent the same relationship between the distance from the start of the trail and the elevation? Explain your reasoning.

A kindergarten teacher bought $21 worth of stickers and cardstock for his class. The stickers cost $1.50 a sheet and the cardstock cost $3.50 per pack. The equation  represents the relationship between sheets of stickers, , packs of cardstocks, , and the dollar amount that the kindergarten teacher spent on these supplies. 

1. Explain how we can tell that this graph represents the given equation.

   

2. What do the vertical and horizontal intercepts, and , mean in this situation?

In physics, the equation is called the ideal gas law. It is used to approximate the behavior of many gases under different conditions.

, , and  represent pressure, volume, and temperature,  represents the number of moles of gas, and is a constant for the ideal gas.

Which equation is solved for ?

To raise funds for uniforms and travel expenses, the soccer team is holding a car wash in a part of town with a lot of car and truck traffic. The team spent $90 on supplies like sponges and soap. They plan to charge $10 per car and $20 per truck. Their goal is to raise $460.

How many cars do they have to wash if they wash the following numbers of trucks?

1. 4 trucks
2. 15 trucks
3. 21 trucks
4. 27 trucks
5.  trucks

During the Middle Ages, people often used grains, scruples, and drahms to measure the weights of different medicines.

If 120 grains are equivalent to 6 scruples and 6 scruples are equivalent to 2 drahms, how many drahms are equivalent to 300 grains? Explain your reasoning. If you get stuck, try creating a table.

Explain why the equation has no solutions.

Consider the equation . If we solve this equation for , which equation would result? 

Diego is buying shrimp and rice to make dinner. Shrimp costs $6.20 per pound and rice costs $1.25 per pound. Diego spent $10.55 buying shrimp and rice. The relationship between pounds of shrimp , pounds of rice , and the total cost is represented by the equation .

Write an equation that makes it easy to find the number of pounds of rice if we know the number of pounds of shrimp purchased.

Here are 2 polygons:

Select all sequences of translations, rotations, and reflections below that would take polygon to polygon .