<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are reading an older version of this FlexBook® textbook: CK-12 Algebra I - Second Edition Go to the latest version.

# 1.3: Patterns and Equations

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Write an equation.
• Use a verbal model to write an equation.
• Solve problems using equations.

## Introduction

In mathematics, and especially in algebra, we look for patterns in the numbers that we see. The tools of algebra assist us in describing these patterns with words and with equations (formulas or functions). An equation is a mathematical recipe that gives the value of one variable in terms of the other.

For example, if a theme park charges $12 admission, then the number of people who enter the park every day and the amount of money taken by the ticket office are related mathematically. We can write a rule to find the amount of money taken by the ticket office. In words, we might say “The money taken in dollars is (equals) twelve times the number of people who enter the park.” We could also make a table. The following table relates the number of people who visit the park and the total money taken by the ticket office. $& \text{Number of visitors} & & 1 & & 2 & & 3 & & 4 & & 5 & & 6 & & 7 \\& \text{Money taken} (\) & & 12 & & 24 & & 36 & & 48 & & 60 & & 72 & & 84$ Clearly, we will need a big table if we are going to be able to cope with a busy day in the middle of a school vacation! A third way we might relate the two quantities (visitors and money) is with a graph. If we plot the money taken on the vertical axis and the number of visitors on the horizontal axis, then we would have a graph that looks like the one shown as follows. Note that this graph shows a smooth line for non-whole number values of $x$ (e.g., $x=2.5$). But, in real life this would not be possible because you cannot have half a person enter the park. This is an issue of domain and range, something we will talk about in the following text. The method we will examine in detail in this lesson is closer to the first way we chose to describe the relationship. In words we said that “The money taken in dollars is twelve times the number of people who enter the park.” In mathematical terms we can describe this sort of relationship with variables. A variable is a letter used to represent an unknown quantity. We can see the beginning of a mathematical formula in the words. The money taken in dollars is twelve times the number of people who enter the park. This can be translated to: the money taken in dollars $= 12 \ \times$ (the number of people who enter the park) To make the quantities more visible they have been placed in parentheses. We can now see which quantities can be assigned to letters. First we must state which letters (or variables) relate to which quantities. We call this defining the variables: Let $x =$ the number of people who enter the theme park. Let $y =$ the total amount of money taken at the ticket office. We can now show the fourth way to describe the relationship, with our algebraic equation. $y=12x$ Writing a mathematical equation using variables is very convenient. You can perform all of the operations necessary to solve this problem without having to write out the known and unknown quantities in long hand over and over again. At the end of the problem, we just need to remember which quantities $x$ and $y$ represent. ## Write an Equation An equation is a term used to describe a collection of numbers and variables related through mathematical operators. An algebraic equation will contain letters that relate to real quantities or to numbers that represent values for real quantities. If, for example, we wanted to use the algebraic equation in the example above to find the money taken for a certain number of visitors, we would substitute that value in for $x$ and then solve the resulting equation for $y$. Example 1 A theme park charges$12 entry to visitors. Find the money taken if 1296 people visit the park.

Let’s break the solution to this problem down into a number of steps. This will help us solve all the problems in this lesson.

Step 1 Extract the important information.

$(\text{money taken in dollars}) & = 12 \times (\text{number of visitors})\\ (\text{number of visitors}) & = 1296$

Step 2 Translate into a mathematical equation.

We do this by defining variables and by substituting in known values.

$\text{Let} \ y & = (\text{money taken in dollars})\\ y & = 12 \times 1296 & & \text{THIS IS OUR EQUATION}.$

Step 3 Solve the equation.

$y=15552 & & \text{Answer: The money taken is} \ \15552$

Step 4 Check the result.

If $15552 is taken at the ticket office and tickets are$12, then we can divide the total amount of money collected by the price per individual ticket.

$\text{(number of people)} = \frac{15552} {12} = 1296$

Our answer equals the number of people who entered the park. Therefore, the answer checks out.

Example 2

The following table shows the relationship between two quantities. First, write an equation that describes the relationship. Then, find out the value of $b$ when $a$ is 750.

$& a: & & 0 & & 10 & & 20 & & 30 & & 40 & & 50 \\& b: & & 20 & & 40 & & 60 & & 80 & & 100 & & 120$

Step 1 Extract the important information. We can see from the table that every time $a$ increases by 10, $b$ increases by 20. However, $b$ is not simply twice the value of $a$. We can see that when $a=0, b=20$ so this gives a clue as to what rule the pattern follows. Hopefully you should see that the rule linking $a$ and $b.$

“To find $a$, double the value of $a$ and add 20.”

Step 2 Translate into a mathematical equation:

Text Translates to Mathematical Expression
“To find $b$ $\rightarrow$ $b=$
“double the value of $a$ $\rightarrow$ $2a$
“add 20” $\rightarrow$ +20

$b=2a+20 & & \text{THIS IS OUR EQUATION}.$

Step 3 Solve the equation.

Go back to the original problem. We substitute the values we have for our known variable and rewrite the equation.

$â€œ\text{when} \ a \ \text{is} \ 750â€ && \rightarrow && b=2(750)+20$

Follow the order of operations to solve

$b& =2(750)+20\\b& =1500+20=1520$

Step 4 Check the result.

In some cases you can check the result by plugging it back into the original equation. Other times you must simply double-check your math. Double-checking is always advisable. In this case, we can plug our answer for $b$ into the equation, along with the value for $a$ and see what comes out. $1520=2(750)+20$ is TRUE because both sides of the equation are equal and balance. A true statement means that the answer checks out.

## Use a Verbal Model to Write an Equation

In the last example we developed a rule, written in words, as a way to develop an algebraic equation. We will develop this further in the next few examples.

Example 3

The following table shows the values of two related quantities. Write an equation that describes the relationship mathematically.

$x-$value $y-$value
$-2$ 10
0 0
2 -10
4 -20
6 -30

Step 1 Extract the important information.

We can see from the table that $y$ is five times bigger than $x$. The value for $y$ is negative when $x$ is positive, and it is positive when $x$ is negative. Here is the rule that links $x$ and $y$.

$y$ is the negative of five times the value of $x$

Step 2 Translate this statement into a mathematical equation.

Text Translates to Mathematical Expression
$y$ is” $\rightarrow$ $y=$
“negative 5 times the value of $x$ $\rightarrow$ $-5x$

$y=-5x && \text{THIS IS OUR EQUATION}.$

Step 3 There is nothing in this problem to solve for. We can move to Step 4.

Step 4 Check the result.

In this case, the way we would check our answer is to use the equation to generate our own $xy$ pairs. If they match the values in the table, then we know our equation is correct. We will substitute $x$ values of -2, 0, 2, 4, 6 in and solve for $y$.

$&x=-2: && y=-5(-2) && y=+10\\&x=0:&& y=-5(0)&&y=0\\&x=2:&& y=-5(2)&&y=-10\\&x=4:&& y=-5(4)&&y=-20\\&x=6:&& y=-5(6)&&y=-30$

Each $xy$ pair above exactly matches the corresponding row in the table.

The answer checks out.

Example 4

Zarina has a $100 gift card, and she has been spending money on the card in small regular amounts. She checks the balance on the card weekly, and records the balance in the following table. Week Number Balance ($)
1 100
2 78
3 56
4 34

Write an equation for the money remaining on the card in any given week.

Step 1 Extract the important information.

## Review Answers

1. $P=20t; P=$ profit; $t=$ number of days. $P =$ profit; $t =$ number of days
2. Profit = 200
1. $y=24-x; y=$ number of cookies in the jar; $x=$ number of cookies eaten
2. 15 cookies
1. $x=$ the number; $7x=35$; number = 5
2. $x=$ another number; $2x+25=$ another number; $5x+5(2x+25)=350$; numbers = 15 and 55
3. $x =$ first integer; $x+1=$ second integer; $x+x+1=35$ ; first integer = 17, second integer = 18
4. $x=$ Peter’s age; $x=3(x-6)$ ; Peter is 9 years old.
1. 3 liters
2. 1:30 pm
3. \$150

Feb 23, 2012

## Last Modified:

Feb 12, 2015
Files can only be attached to the latest version of None

# Reviews

Please wait...
Please wait...
Image Detail
Sizes: Medium | Original

CK.MAT.ENG.SE.2.Algebra-I.1.3