Identifying the Most Useful Form of an Equation

1.8.2 Identifying the Most Useful Form of an Equation

1.8.2 • Identifying the Most Useful Form of an Equation

Activity

For questions 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.

8 volunteers

Compare your answer:

$\frac{1}{4}$ mile

10 volunteers

Compare your answer:

$\frac{1}{5}$ mile

25 volunteers

Compare your answer:

$\frac{2}{25}$ mile or 0.08 mile

36 volunteers

Compare your answer:

$\frac{2}{36}$ or $\frac{1}{18}$ mile

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

Compare your answer:

$l = \frac{2}{n}$

For questions 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.

0.4 mile

Compare your answer:

$n = \frac{2}{0.4} =$ 5 volunteers

$\frac{2}{7}$ mile

Compare your answer:

$n = \frac{2}{\frac{2}{7}} = \frac{2}{1} \times \frac{7}{2} =$ 7 volunteers

0.125 mile

Compare your answer:

$n = \frac{2}{0.125} =$ 16 volunteers

$\frac{6}{45}$ mile

Compare your answer:

$n = \frac{2}{\frac{6}{45}} = \frac{2}{1} \times \frac{45}{6} =$ 15 volunteers

10.

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

Compare your answer:

$n = \frac{2}{l}$

Are you ready for more?

Extending Your Thinking

Let's think about the graph of the equation $y = \frac{2}{x}$ .

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

Compare your answer:

Your answer may vary, but here is an example:

$(- 2.5, - 0.8)$
$(- 1, - 2)$
$(0.5, 4)$
$(1.25, 1.6)$

Access multimedia content

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

Compare your answer:

$Image of Demos screen of the plots of coordiantes for the equation of the fraction two over x.$ A scatter plot with grid lines, displaying six black dots positioned at various coordinates around the origin on the x and y axes.

A scatter plot with grid lines, displaying six black dots positioned at various coordinates around the origin on the x and y axes.

What do you think the graph looks like when $x$ is between 0 and $\frac{1}{2}$ ? Continue to use the graphing tool or technology outside the course to graph the function $y = \frac{2}{x}$ with your points. Try some values of $x$ to test your idea. (Students were provided access to Desmos.)

Compare your answer:

Your answer may vary, but here is an example:

The graph gets steeper and steeper as $x$ gets closer to 0.

A graph on a Cartesian grid with both vertical and horizontal axes. It features a hyperbola that approaches the axes without touching them.

What is the largest value $y$ can ever be?

$y = \frac{2}{x}$ with your points. Try some values of $x$ to test your idea. In Desmos, zoom in on your graph.

Compare your answer:

Your answer may vary, but here is an example:

There is no limit to the value of 𝑦.

Self Check

Use formula for the area of a triangle,

A=\frac{1}{2}bh

, 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?

$\frac{1}{2}b=\frac{A}{h}$
$2A=bh$
$h=\frac{2A}{b}$
$b=\frac{2A}{h}$

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 $V = \frac{1}{3} π r^{2} h$ 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 $V = \frac{1}{3} π r^{2} h$ for $h$ .

Step 1 - Remove the fraction on the right.

Multiply both sides by 3.

Step 2 - Simplify.

Step 3 - Divide both sides by $π r^{2}$ .

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 $h = \frac{3 V}{π r^{2}}$

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

$C = \frac{5}{9} (F - 32)$ for $F$ .

Step 1 - Remove the fraction on the right.

Multiply both sides by $\frac{9}{5}$ .

Step 2 - Simplify.

Step 3 - Add 32 to both sides.

Example 3

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

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

Step 2 - Simplify.

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

Step 4 - Simplify.

Example 4

$\frac{2 d}{t} = (v_{0} + v)$

Now, try to solve $d = \frac{1}{2} (v_{0} + v) t$ for $v_{0}$ .

Answer:

Step 1 - Remove the fraction on the right.

Multiply by 2.

$2 \times d = 2 \times \frac{1}{2} (v_{0} + v) t$

Step 2 - Simplify.

$2 d = (v_{0} + v) t$

Step 3 - Divide both sides by $t$ .

$\frac{2 d}{t} = \frac{(v_{0} + v) t}{t}$

Step 4 - Simplify.

$\frac{2 d}{t} = (v_{0} + v)$

Step 5 - Subtract $v$ from both sides to solve for $v_{0}$ .

$\frac{2 d}{t} - v = v_{0} + v - v$

$\frac{2 d}{t} - v = v_{0}$

Try it

Solve a Formula for a Specific Variable

Solve $8 x + 7 y = 15$ for $x$ .

Here is how to solve this equation for $x$ .

Step 1 - Subtract $7 y$ from each side.

$8 x + 7 y - 7 y = 15 - 7 y$

Step 2 - Simplify.

$8 x = 15 - 7 y$

Step 3 - Divide both sides by 8.

$\frac{8 x}{8} = \frac{15 - 7 y}{8}$

The equation solved for $x$ is $x = \frac{15}{8} - \frac{7 y}{8}$ .