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Algebra 1

1.8.2 Identifying the Most Useful Form of an Equation

Algebra 11.8.2 Identifying the Most Useful Form of an Equation

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Activity

For numbers 1 - 4, use the situation below:

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 show your reasoning.

Mummers' Parade in Philly.
1.

8 volunteers

2.

10 volunteers

3.

25 volunteers

4.

36 volunteers

5.

Write an equation that would make it easy to find ll, the length of a road section in miles for each volunteer, if there are nn volunteers.

For numbers 6 - 9, find the number of volunteers in the group if each volunteer cleans up a section of the following lengths. Be prepared to show your reasoning.

6.

0.4 mile

7.

2727 mile

8.

0.125 mile

9.

645645 mile

10.

Write an equation that would make it easy to find the number of volunteers, nn, if each volunteer cleans up a section that is ll miles.

Are you ready for more?

Extending Your Thinking

Let's think about the graph of the equation y=2xy=2x.

1.

Create a list (x,y)(x,y) pairs that will help you graph the equation. Make sure to include some negative numbers for xx and some numbers that are not integers.

2.

Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool above. 

3.

What do you think the graph looks like when xx is between 0 and 1212? Continue to use the graphing tool or technology outside the course to graph the function y=2xy=2x with your points. Try some values of xx to test your idea. (Students were provided access to Desmos.)

4.

What is the largest value yy can ever be?

y=2xy=2x with your points. Try some values of xx to test your idea. In Desmos, zoom in on your graph.

Self Check

Use formula for the area of a triangle, A = 1 2 b h , where b is the base of the triangle and h is the height of the triangle. Which equation would best be used to find the base of the triangle?
  1. 1 2 b = A h
  2. 2 A = b h
  3. h = 2 A b
  4. b = 2 A h

Additional Resources

Solve a Formula for a Specific Variable

We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in many fields, so it is important to recognize formulas and be able to manipulate them easily.

It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equal sign with a coefficient of 1, and all other variables and constants are on the other side of the equal sign.

Geometric formulas often need to be solved for another variable, too. The formula V=13πr2hV=13πr2h is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.

Example 1

Solve V=13πr2hV=13πr2h for hh.

Step 1 - Remove the fraction on the right.

Multiply both sides by 3.

Three times V equals three times one third times pi times r squared times h

Step 2 - Simplify.

Three times V equals  pi times r squared times h

Step 3 - Divide both sides by πr2πr2.

Three times V over pi times r squared equals h

We could now use this formula to find the height of a right circular cone when we know the volume and the radius of the base, by using the formula h=3Vπr2h=3Vπr2

In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.

Example 2

Solve

C=59(F32)C=59(F32) for FF.

Step 1 - Remove the fraction on the right.

Multiply both sides by 9/5.

Nine fifths times C equals nine fifths times five ninths open parenthesis F minus thirty two closed parenthesis

Step 2 - Simplify.

Nine fifths times C equals open parenthesis F minus thirty two closed parenthesis

Step 3 - Add 32 to both sides.

Nine fifths times C plus thitry two equals F

Continue explanation as is

Example 3

Solve the formula 8x+7y=158x+7y=15 for yy.

Step 1 - Subtract 8x from both sides to isolate the term with yy.

Eight times x minus eight times x plus seveb equals fifteen minus eight times x

Step 2 - Simplify.

Seven times y equals fifteen minus eight x

Step 3 - Divide both sides by 7 to make the coefficient of yy equal to 1.

Seven times y over seven equals fifteen minus eight x over seven

Step 4 - Simplify.

2dt=(v0+v)2dt=(v0+v)

Now, try to solve d=12(v0+v)td=12(v0+v)t for v0v0.

Answer:

Step 1 - Remove the fraction on the right.

Multiply by 2.

2×d=2×12(v0+v)t2×d=2×12(v0+v)t

Step 2 - Simplify.

2d=(v0+v)t2d=(v0+v)t

Step 3 - Divide both sides by tt.

2dt=(v0+v)tt2dt=(v0+v)tt

Step 4 - Simplify.

2dt=(v0+v)2dt=(v0+v)

Step 5 - Subtract vv from both sides to solve for v0.

2dtv=v0+vv2dtv=v0+vv

2dtv=v02dtv=v0

Answer:

Try it

Try It: Solve a Formula for a Specific Variable

Solve 8x+7y=158x+7y=15 for xx.

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