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Mathematics can be used to model real-life situations. This is helpful to understand those situations and find solutions to problems that might arise in such scenarios. This lesson will discuss the concepts of *variable*, *linear expression*, *equation*, and *linear equation*.
### Catch-Up and Review

**Here is a recommended reading before getting started with this lesson.**

Challenge

A group of people at a restaurant is waiting for a table for a family dinner. Waiters began adding tables one by one until the entire group was seated as requested.

Is there an expression to represent the number of people in terms of the number of tables?

Discussion

Sometimes a quantity is unknown or its value may change. If this is the case, the best way to represent this quantity is using a *variable*.

A variable is a symbol used to represent an unknown quantity. Often, variables represent fixed but unknown numbers. Variables are usually denoted with letters such as $x.$

$x+1=8 $

In this case, $x$ is a variable in an equation. Solving this equation for $x$ will identify the value of $x$ that makes the equation true.
$x+1=8⇔x=7 $

Alternatively, a variable can be used to represent a quantity that changes.
$Izabella’s income varies depending on thenumber of hours she works.She receives afixed salary of$100per weekplus$4per hour. $

In this case, Izabella's weekly income could be different every week, depending on the number of hours she works. Therefore, the use of a variable is appropriate. Let $x$ be the hours that Izabella works in a week. Her income can be written by adding the fixed $$100$ to the $$4$ per hour. $Izabella’s Weekly Income4x+100 $

Discussion

There are many real-life situations that can be modeled by combining numbers and variables in algebraic expressions. A special case of algebraic expressions are *linear expressions*.

Concept

A linear term is an algebraic expression that includes a coefficient multiplied by a variable with an exponent of one. A linear expression is an expression that includes at least one linear term and any constant terms. No other type of terms may be included. The most common form of a linear expression is given below.

In this expression, $a$ and $b$ are real numbers, with $a =0.$ To completely understand the definition of a linear expression, some important concepts will be be broken down. Consider the example linear expression $x−5y+2.$

$x−5y+2$ | ||
---|---|---|

Concept | Explanation | Example |

Term | Parts of an expression separated by a $+$or $−$sign. |
$x,$ $-5y,$ $2$ |

Coefficient | A constant that multiplies a variable. If a coefficient is $1,$ it does not need to be written due to the Identity Property of Multiplication. | $1,$ $-5$ |

Linear Term | A term that contains exactly one variable whose exponent is $1.$ | $x,$ $-5y$ |

Constant Term | A term that contains no variables. It consists only of a number with its corresponding sign. | $2$ |

The following table shows some examples of linear and non-linear expressions.

Linear Expressions | Non-linear Expressions |
---|---|

$3x$ | $5$ |

$-5y+1$ | $2xy−3$ |

$3x−21 y+2$ | $x1 −2$ |

$πx+6y$ | $5x_{2}+x−1$ |

$3x+4−2x+9−x=13 $

The expression $3x+4−2x+9−x$ looks like a linear expression, but the result after simplifying is $13,$ which is a constant and therefore not a linear expression.Example

Ramsha, Tiffaniqua, Zosia, and some of their friends are spending the summer at math camp. One of their first lessons is about identifying parts of linear expressions. They are given the following table to fill in based on the linear expressions in the top row.

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | |||

$x-$term | |||

$y-$term | |||

Linear Term(s) | |||

Coefficient(s) | |||

Constant Term |

Help them fill in the table.

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | $6x$ and $-1$ | $2 $ and $x$ | $-2x,$ $3,$ and $-9y$ |

$x-$term | $6x$ | $x$ | $-2x$ |

$y-$term | - | - | $-9y$ |

Linear Term(s) | $6x$ | $x$ | $-2x$ and $-9y$ |

Coefficient(s) | $6$ | $1$ | $-2$ and $-9$ |

Constant Term | $-1$ | $2 $ | $3$ |

When there is no number in front of a variable, the coefficient is $1.$ The $x-$ and $y-$terms are the terms that contain the variables $x$ and $y,$ respectively.

Start by finding the terms of each expression. Recall that the terms are the parts separated by addition or subtraction signs.

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | $6x$ and $-1$ | $2 $ and $x$ | $-2x,$ $3,$ and $-9y$ |

$x-$term | |||

$y-$term | |||

Linear Term(s) | |||

Coefficient(s) | |||

Constant Term |

The $x-$term is the term that contains the variable $x.$

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | $6x$ and $-1$ | $2 $ and $x$ | $-2x,$ $3,$ and $-9y$ |

$x-$term | $6x$ | $x$ | $-2x$ |

$y-$term | |||

Linear Term(s) | |||

Coefficient(s) | |||

Constant Term |

Similarly, the $y-$term is the term that contains the variable $y.$ In this case, the first two expressions do not have a $y-$term.

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | $6x$ and $-1$ | $2 $ and $x$ | $-2x,$ $3,$ and $-9y$ |

$x-$term | $6x$ | $x$ | $-2x$ |

$y-$term | - | - | $-9y$ |

Linear Term(s) | |||

Coefficient(s) | |||

Constant Term |

The linear terms are the terms that contain only one variable, raised to the power of $1.$

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | $6x$ and $-1$ | $2 $ and $x$ | $-2x,$ $3,$ and $-9y$ |

$x-$term | $6x$ | $x$ | $-2x$ |

$y-$term | - | - | $-9y$ |

Linear Term(s) | $6x$ | $x$ | $-2x$ and $-9y$ |

Coefficient(s) | |||

Constant Term |

Next, the coefficients are the numbers that multiply a variable. When there is no number in front of a variable, the coefficient is $1.$

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | $6x$ and $-1$ | $2 $ and $x$ | $-2x,$ $3,$ and $-9y$ |

$x-$term | $6x$ | $x$ | $-2x$ |

$y-$term | - | - | $-9y$ |

Linear Term(s) | $6x$ | $x$ | $-2x$ and $-9y$ |

Coefficient(s) | $6$ | $1$ | $-2$ and $-9$ |

Constant Term |

Finally, the constant term is any term without a variable.

$6x−1$ | $2 +x$ | $-2x+3−9y$ | |
---|---|---|---|

Term(s) | $6x$ and $-1$ | $2 $ and $x$ | $-2x,$ $3,$ and $-9y$ |

$x-$term | $6x$ | $x$ | $-2x$ |

$y-$term | - | - | $-9y$ |

Linear Term(s) | $6x$ | $x$ | $-2x$ and $-9y$ |

Coefficient(s) | $6$ | $1$ | $-2$ and $-9$ |

Constant Term | $-1$ | $2 $ | $3$ |

Pop Quiz

Select the required parts of the given linear expressions.

Example

Ramsha is on the second day of a hiking excursion at math camp. On the first day, she hiked from base camp to the first station.

Her current distance from base camp, in miles, is given by the following linear expression.$Distance From the Base Camp1.5x+10 $

Here, $x$ is the number of hours Ramsha walks on the second day. a What does the coefficient $1.5$ represent in this context?

b What does the linear term $1.5x$ represent in this context?

c What does the constant term $10$ represent in this context?

a The speed at which Ramsha walks.

b The distance Ramsha walks on the second day.

c The distance from base camp to the first station.

b The expression represents distance. Therefore, each term in the expression also represents distance.

c The constant term represents a fixed distance.

a Start by representing Ramsha's current position from the base camp in the diagram.

Note that $1.5x+10$ represents a distance and the units of each term are miles. However, the units of $x$ are *hours*. This gives a clue of the units of the coefficient $1.5.$

In order to get rid of hours

and end up with miles,

the units of the coefficient must be miles per hour.

Therefore, the coefficient represents speed, or more precisely, the speed at which Ramsha walks.

b Since $x$ represents the number of hours Ramsha has walked on the *second day,* then $1.5x$ represents the distance Ramsha has walked on the second day, starting at the first station.

c Finally, since $1.5x+10$ represents the distance from base camp and $1.5x$ the distance walked on the second day, then the constant term $10$ represents the distance Ramsha walked on the first day — that is, the distance from base camp to the first station.

Discussion

In the previous example, an expression modeling a real-life scenario was provided. However, most of the time such expressions are not provided and must instead be created. The use of *inductive reasoning* can help with this process.

Inductive reasoning is the process of finding patterns in specific observations and writing a conclusion or conjecture. Since the conjecture is based on observations, it might be false. For example, suppose an observer notices that all the birds around them are white. The observer might inductively reason that all birds in the world are white, which is not true.

Inductive reasoning is a practical method used in geometry and algebra for recognizing visual and numerical patterns. For instance, use the first three figures in the following diagram to guess the shape and the number of cubes in the fourth figure.

After observing the first three figures, it is reasonable to conclude that the number of cubes increases by $3$ from one figure to the next — one on the top, one on the front, and one on the right. Since the figure starts with $1$ cube and each step adds $3$ cubes, an expression can be written to model the number of cubes in each figure.

$1+3n $

Here, $n$ represents the number of steps after Figure $1.$ For example, Figure $2$ is one step after Figure $1,$ so $n=1.$ This expression allows the calculation of the number of cubes in any figure of the pattern. $n$ | Number of Cubes | |
---|---|---|

Figure $1$ | $0$ | $1+3⋅0=1$ |

Figure $2$ | $1$ | $1+3⋅1=4$ |

Figure $3$ | $2$ | $1+3⋅2=7$ |

Figure $4$ | $3$ | $1+3⋅3=10$ |

Figure $121$ | $120$ | $1+3⋅120=361$ |

Example

At the end of the math camp, the students all visit an amusement park. Tiffaniqua did so well at camp that she won a coupon that allows her entire friend group to go on rides for $x$ dollars per ride. By the end of the day, the group has gone on $11$ rides and each person has eaten one hot dog.

The price of a hot dog is $d$ dollars and the group is formed by $k$ people, including Tiffaniqua.Write an expression that represents the total amount of money the group paid.

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The total amount of money paid is the sum of the amount paid for the $11$ rides and the amount spent in hot dogs. The cost per person is the total cost divided by the number of people.

Start by writing what each letter represents.

$xdk →cost of a ride→cost of a hot dog→number of people $

Thanks to the coupon, the entire group pays $x$ dollars per ride. Since they went on $11$ rides, they paid $11x$ dollars.
$Cost of the rides:11x $

The cost of the hot dogs must be added to the previous amount. The price of a hot dog is $d$ dollars. Since there are $k$ people, the total cost is $kd$ dollars.
$Cost of the hot dogs:kd $

Finally, the expression representing the total amount the group paid is the sum of the amount paid in rides and the amount paid in hot dogs.
$Total cost:11x+kd $

If the group wants to divide the cost evenly, then the total cost must be divided by the total number of people. In this case, the expression above must be divide by $k.$
$Cost per person:k11x+kd $

Discussion

In the previous two examples, each situation was represented by an algebraic expression. There are some cases where writing an expression is not enough to model a situation. In such cases, an *equation* might be required.

Concept

An equation is a type of mathematical relation that indicates that two quantities are equal. Equations often contain one or more unknowns values called variables. Some examples are shown below.

Solving an equation with a variable means to find the value or values of each variable that make the equation true.

Discussion

Equations always contain an equals sign, while expressions do not. In fact, an equation can be seen like a statement that connects two expressions with an equals sign.

When learning about equations, a good first step is to start with *linear equations*, which involve only one variable.

Concept

A linear equation is an equation with at least one linear term and any number of constants. No other types of terms may be included. Linear equations in one variable have the following form, where $a$ and $b$ are real numbers and $a =0.$

$ax+b=0orax=b $

Example

Zosia wants to go shopping after camp, but she realizes that she spent too much money at the amusement park. She decides to start saving money and go shopping when she has $$25$ saved up. She puts $$15$ in her piggy bank and plans to add $$0.50$ every day.
### Hint

### Solution

Let $t$ be the number of days Zosia has been saving money. Write an equation that models the moment when her savings reaches $$25.$

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Zosia's savings can be modeled by a linear expression of the form $at+b,$ where $a$ and $b$ are real numbers.

First, write an expression modeling Zosia's savings. To do this, note that the amount of money in the piggy bank increases by a constant amount every day. Therefore, it can be modeled by a linear expression.

$at+b $

The coefficient $a$ is the rate of change and the constant $b$ is the initial amount. Since Zosia starts with $$15$ and saves $$0.50$ every day, it can be said that $a=0.5$ and $b=15.$
$0.5t+15 $

The expression above represents the money in the piggy bank after $t$ days. Finally, the expression above has to be equated to something else.
$0.5t+15=? $

To figure out what comes on the right-hand side, translate the equation above into words.
$The money in the piggy bank aftertdays is equal to… $

Since the aim is to model the moment in which Zosia reaches $$25,$ the right-hand side of the equation is $25.$ $0.5t+15=25$

Example

Now that she has money saved up, Zosia is finally going to the mall. At the candy store, she buys $3$ bags containing the same number of candies. The store also gives her $2$ free candy samples. When she arrives home, she opens the bags and counts a total of $17$ candies.
### Hint

### Solution

Let $x$ be the number of candies in each bag. Write an equation that models the total amount of candies Zosia got from the store.

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What is the coefficient for the $x-$term in this situation?

First, write an expression that models the number of candies from the store. To do so, note that the total number of candies depends on the number of candies in each bag, the number of bags, and the number of extra candies. A linear expression can be used.

$ax+b $

The variable $x$ represents the number of candies in each bag. The coefficient $a$ is the number of bags, and the constant $b$ is the number of free candies. Then, $a=3$ and $b=2.$
$3x+2 $

The expression above represents the total number of candies after opening the $3$ bags with $x$ candies each. Since an equation is required, the expression above has to be equated to something else.
$3x+2=? $

Zosia counted $17$ total candies. Therefore, the right-hand side of the equation is $17.$
$3x+2=17 $

Closure

This lesson discussed linear expressions and linear equations, as well as how to use inductive reasoning to create them. Countless real-life situations can be modeled by algebraic expressions and equations. Consider again the challenge presented at the beginning of the lesson. ### Hint

### Solution

As the waiters add more tables for the group of people at the restaurant for the family dinner, they begin to notice a pattern. Help the waiters by writing an algebraic expression that represents the number of people they can seat in terms of the number of tables $x.$

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Notice how the number of people sat increases with each table added.

Begin by making sense of the situation. At least $1$ table is needed for the family dinner. In this case, $4$ people can be seated.

For each subsequent table, $two$ more people are seated.

A linear expression can be used to model this situation.$ax+b $

Here, $a=2$ because adding a table will add two more seated people. It can be noted that there will be always $2$ people $at$ $the$ $ends$ of the big table.
Therefore, it can be said that $b=2.$ With this, the expression is complete.

$Number of people=2x+2 $

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