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After a parade, a group of volunteers is helping to pick up the trash along a 2-mile stretch of a road.
The group decides to divide the length of the road so that each volunteer is responsible for cleaning up equal-length sections.
Find the length of a road section for each volunteer if there are the following numbers of volunteers. Be prepared to explain or show your reasoning.
Find the number of volunteers in the group if each volunteer cleans up a section of the following lengths. Be prepared to explain or show your reasoning.
A relationship between quantities can be described in more than one way. Some ways are more helpful than others, depending on what we want to find out. Let’s look at the angles of an isosceles triangle, for example.
The two angles near the horizontal side have equal measurement in degrees, .
The sum of angles in a triangle is , so the relationship between the angles can be expressed as:
Suppose we want to find when is .
Let's substitute 20 for and solve the equation.
What is the value of if is ?
Now suppose the bottom two angles are each. How many degrees is the top angle?
Let's substitute 34 for and solve the equation.
What is the value of if is ?
Notice that when is given, we did the same calculation repeatedly to find : We substituted into the first equation, subtracted from 180, and then divided the result by 2.
Instead of taking these steps over and over whenever we know and want to find , we can rearrange the equation to isolate :
This equation is equivalent to the first one. To find , we can now simply substitute any value of into this equation and evaluate the expression on the right side.
Likewise, we can write an equivalent equation to make it easier to find when we know :
Rearranging an equation to isolate one variable is called solving for a variable. In this example, we have solved for and for . All three equations are equivalent. Depending on what information we have and what we are interested in, we can choose a particular equation to use.