3.8: Solving more Equations

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INTRODUCTION

As you keep taking more advanced math courses in the future, you will keep seeing more operations, procedures, or functions that cancel each other out. Many operations, procedures, or functions come in pairs. Whatever one operation does, the other one will undo.

By now, you are fairly familiar with two pairs of such operations:

Addition and subtraction
Multiplication and division

In this collaboration, we will be discussing two more pairs of operations that cancel each other out:

Squaring a number (or simply multiplying it by itself two times) and finding a square root.
Cubing a number (or simply multiplying it by itself three times) and finding a cube root.

Similar to what we did in Collaboration 3.6, let’s illustrate it with an example:

Pick any non-negative number. This example will use 11.
Next, square the number that you picked. In this example, 11² = 121.
Next, square root the number for the previous step. You can do it by pressing the √ key on your calculator. In my case, $\sqrt{121}$ = 11. Note: In this collaboration, all examples involve positive numbers, which is why we disregard the negative value.
11 is the same number that I started with! This shows how squaring a number and taking a square root of a number cancel each other out.

This is not a coincidence. It works for any numbers and variables. For example,

$\sqrt{71^{2}}$ = 71
$\sqrt{x^{2}}$ = x

The same applies to cubes and cube roots. Let’s illustrate with an example:

Pick any number. This example will use 8.
Next, cube the number that you picked. In my case, 8³ = 512.
Next, cube root the number for the previous step. You can do it by pressing the $\sqrt[3]{x}$ key on your calculator. In my case, $\sqrt[3]{512}$ = 8.
8 is the same number that I started with and so did you! This shows how cubing numbers and taking a cube root of a number cancel each other out.

This is not a coincidence. It works for any numbers and variables. For example,

$\sqrt[3]{87.5^{3}}$ = 87.5
$\sqrt[3]{T^{3}}$ = T

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

To solve any equation, one must follow the basic rules of undoing and keeping the equation balanced.

By the end of this collaboration, you should be able to

Solve linear equations that require simplification before solving.
Solve for a variable in a linear equation in terms of another variable.
Solve for a variable in a single-term quadratic equation.

PROBLEM SITUATION: SOLVING EQUATIONS OF DIFFERENT FORMS

Solving equations such as the Widmark equation for blood alcohol content (BAC) and proportional equations for resizing graphics is an important skill. Mathematical models are often constructed to represent real-life situations. Being able to use these equations fully includes being able to solve for various unknown variables in the equation. Below are three scenarios to help you practice and enhance your equation-solving skills. With each answer, check that it is reasonable given the context and check that you have included the correct units with your solution.

(1) Paula has two options for going to school. She can carpool with a friend or take the bus. Her friend estimates that driving will cost 22 cents per mile for gas and 8.2 cents per mile for maintenance of the car. Additionally, there is a $25 parking fee per week at the college. If Paula carpools, she would pay half of these costs. The cost of the carpool can be modeled by the following equation where C is the cost of carpooling per week and m is the total miles driven to school each week:

\[C = \frac{1}{2}(0.082m + 0.22m + 25)\nonumber \]

(a) On your own, take a minute to think about what the 12 represents in the equation above.

(b) In your group, discuss what each term inside the parentheses represents.

(c) In the given formula, 0.082m is the total maintenance costs for m miles and 0.22m is the total gas cost for m miles. Since both terms include the same variable with the same exponent, they can be combined into one term by adding (or subtracting) their coefficients. Discuss in your group how to rewrite the given formula by combining like-terms. In other words, rewrite the given formula to only include one term that represents the cost per mile.

(d) Find the total weekly carpooling cost if the commute to school is seven miles each way and Paula goes to school three times a week. First try this on your own, then discuss as a group when everyone is ready. Round to the nearest cent.

(e) A weekly bus pass costs $22.00 dollars. How many total miles must Paula commute to school each week for the carpool cost to be equal to the bus pass? First try this on your own, then discuss as a group when everyone is ready. Round to the nearest hundredth.

(f) How many trips to school each week must Paula make for the bus pass to be less expensive than carpooling? First try this on your own, then discuss as a group when everyone is ready.

(2) Recall Widmark’s equation for BAC. In the case of the average male who weighs 190 pounds,²⁴ you can simplify Widmark’s formula to get

\[B = −0.015t + 0.022N\nonumber \]

Forensic scientists often use this equation at the time of an accident to determine how many drinks someone had. In these cases, time (t) and BAC (B) are known from the police report. The crime lab uses this equation to estimate the number of drinks (N).

(a) Find the number of drinks if the BAC is 0.09 and the time is two hours. First try this on your own, then discuss as a group when everyone is ready. Round to the nearest drink.

(b) Since forensic scientists use the formula to solve for N, it is easier if the formula is rewritten so that it actually solves for N. In the space below, solve for N in terms of t and B. In other words, rewrite the equation so N is isolated on one side of the equation and all other terms are on the other side. First try this on your own, then discuss as a group when everyone is ready.

(c) Use the new formula to find the number of drinks if the BAC is 0.17 and the time is 1.5 hours. First try this on your own, then discuss as a group when everyone is ready. Round to the nearest drink.

(3) You volunteer for a nonprofit organization interested in women’s issues. The logo for your nonprofit organization is three identical squares arranged as follows:

Image showing the three identical squares, one next to the other, staggered.

(a) The organization wants to make banners of different sizes. Find an equation that can be used to find the total area of the logo based on the length of the side of one of the squares. First try this on your own, then discuss as a group when everyone is ready.

Assume that the 3 squares have the same dimensions.

(b) The organization is sponsoring a walk-a-thon to raise funds for breast cancer research. You want to recreate this logo (forming the three squares) in the middle of the racetrack with bras that have been collected at multiple drop-off sites around the city. You estimate that approximately 1,500 square feet of bras have been donated. How long should you make each side of the square? Round to two decimal places.

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

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²⁴ http://www.cdc.gov/nchs/fastats/body-measurements.htm

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