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Numerical expressions deal with combinations of mathematical operations with numbers, which are known quantities. However, sometimes it is necessary to deal with unknown quantities. This lesson will examine how to incorporate unknown quantities in expressions and how to evaluate that expression once the quantity is known.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

Tearrik is really excited about a game that is coming out this weekend. He decides to sell some of his stuff so that he can make enough money to buy the game.

Proud of him for making efforts to buy his own things, Tearrik's parents decided to double the money he makes and add $$5$ extra on top. How can this situation be written in a mathematical way without knowing how much money Tearrik makes from the garage sale?Discussion

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 $

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

When dealing with variables, sometimes other numbers are needed to complete a mathematical idea. One type of these numbers is *coefficients*.

Theory

$5x $

In this case, the coefficient is $5.$ The coefficient is the multiplier, even if the variable is raised to some power.
$2x_{3} $

In this case, the coefficient is $2.$ Note that a variable with a coefficient of $1$ is usually written without a coefficient. $1x_{2}=x_{2} $

Discussion

Another type of numbers that appear with variables is called *constants*.

Concept

$x+15 $

In this case, $15$ is a constant. It should be noted that not every constant is written with digits. Some special constants are written with special symbols, such as the number pi, which is often written as $π.$Pop Quiz

The applet below displays different variables multiplied by coefficients. Answer the indicated question correctly.

Discussion

An algebraic expression is a valid combination of numbers, variables, and mathematical operations. For example, in the expression $2x+3,$ the variable $x$ is being *multiplied* by its coefficient $2,$ and this product is then *added* to the constant $3.$

Discussion

Algebraic expressions are made by adding or subtracting smaller expressions called *terms*.

Concept

A term is an algebraic or numeric expression that does not involve addition or subtraction. An expression contains one or more terms, separated from one another by

$+$or

$−$signs.

Mathematical Expression | Number of Terms | Terms |
---|---|---|

$7x$ | $1$ | $7x$ |

$8$ | $1$ | $8$ |

$8x−2(5)$ | $2$ | $8x$ and $-2(5)$ |

$x_{2}+y_{2}+4$ | $3$ | $x_{2},$ $y_{2},$ and $4$ |

$2x_{2}−5x−122$ | $3$ | $2x_{2},$ $-5x,$ and $-122$ |

Example

At his garage sale, Tearrik is selling some of his old shirts and pants.

He is selling the pants for $$1$ more than three times the price of a shirt. If the price of a shirt is $s,$ write an algebraic expression for the price of a pair of pants.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["s"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3s+1","3*s+1"]}}

Remember that an algebraic expression is a combination of numbers, variables, and mathematical operations.

An algebraic expression is a combination of numbers, variables, and mathematical operations. The terms are separated by $+$

or $−$

signs. In verbal expressions, some words or phrases may imply certain math operations.

Key Words and Phrases | |
---|---|

Addition | added to, plus, sum of, more than, increased by, total of and |

Subtraction | subtracted from, minus, difference of, less than, decreased by, fewer than, take away |

Multiplication | multiplied by, times, product of, twice |

Division | divided by, quotient of |

$Tearrik is selling the pants at$1more thanthree times the price of a shirt. $

It is time to identify the keywords in this sentence. The phrase $more$ $than$ indicates an addition. In this case, $$1$ is going to be added to some other quantity.
$1+ $

The phrase $three$ $times$ represents a multiplication. In this case, it is $3$ times the price of a shirt, which is $s.$
$1+3s $

There is no more information to include, so this expression represents the price of a pair of pants.
$1+3s $

Example

Tearrik was able to buy the video game he desired and he started playing right away.

In the game, timed challenges reward players with bonus points for finishing them quickly. Confused about how the bonus points are rewarded, Tearrik asked his friend Magdalena about it. She told him how the bonus points work.

Help Tearrik write the information that Magdalena gave him as an algebraic expression. Let the variable be $t.${"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["t"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["1\/2*(400-t)","\\dfrac{1}{2}(400-t)"]}}

Identify the variable. Then, look for keywords in Magdalena's information that indicate operations.

It is important to identify the variables before writing an algebraic expression. In Magdalena's description, the time it takes to finish the challenge can be different for different tries or different people. Since this time can change, it can be assigned to the given variable $t.$

Examining Magdalena's explanation, it is possible to find some of these keywords.

$Time to Finish the Challenge:t $

An algebraic expression is a combination of numbers, variables, and mathematical operations. The terms are separated by $+$or

$−$signs. In verbal expressions, there are words or phrases that indicate certain mathematical operations.

Key Words and Phrases | |
---|---|

Addition | added to, plus, sum of, more than, increased by, total of and |

Subtraction | subtracted from, minus, difference of, less than, decreased by, fewer than, take away |

Multiplication | multiplied by, times, product of, twice |

Division | divided by, quotient of |

$The bonus points arehalfthedifferencebetween400andthe time it took to finish the challenge $

The $half$ indicates a division by $2$ and the $difference$ indicates a subtraction. Magdalena says to take half the difference. Because of this, it is convenient to write the difference first. This difference can be written by subtracting the time $t$ from $400.$
$400−t $

The division by $2$ affects the result of the subtraction. Considering the order of operations, division operations are evaluated before subtraction. To evaluate the subtraction first, it is important to write the subtraction between parentheses.
$21 (400−t) $

This expression can help Tearrik to determine how many bonus points he will get for however much time he takes to solve the challenge.
Discussion

Evaluating an expression consists of determining the value of an expression when the variable or variables of the expression take a specific value. This is done by substituting the given value for the variable in question into the expression and then simplifying it. As an example, consider the following expression.
*expand_more*
*expand_more*
If an expression has more than one variable, each variable is replaced by its given value before all the required operations are performed.

$2(x−1)_{2} $

This expression is going to be evaluated considering that $x$ is equal to $5.$ There are two steps to follow.
1

Substitute the Given Value for $x$

In the expression, substitute the given value for every instance of the variable. In this case, substitute $5$ for $x.$
After the substitution, the variable disappeared and the algebraic expression turned into a numeric expression.

2

Simplify the Expression

Now the expression can be simplified following the order of operations.
Consequently, when $x$ is equal to $5,$ the given expression equals $8.$

$2(5−1)_{2} $

SubTerms

Subtract terms

$24_{2} $

CalcPow

Calculate power

$216 $

CalcQuot

Calculate quotient

$8$

Example

Tearrik continues to enjoy playing his new game. He is focusing on collecting the challenge stars on each level.

a Tearrik wants to collect all the stars in the game. There are $3$ special stars per level. If $x$ is the number of levels, write an expression for the numbers of stars.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3x"]}}

b There are $73$ levels in the game. Use the expression from Part A to find the total number of stars.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"stars","answer":{"text":["219"]}}

a The number of stars per level indicates a multiplication.

b Substitute the given value for the variable and evaluate the expression.

a The total number of bonus stars can be found by adding the number of stars on every level. Every level has $3$ stars. Adding the number of stars of every level is the same as multiplying $3$ by the number of levels, $x.$

$x3+3+…+3 =3x $

b There are $73$ levels in the game, meaning that $x$ is $73.$

$x=73 $

Use this fact and evaluate the algebraic expression from Part A to find the total number of stars in the game. Notice that substituting the value for the variable changes the algebraic expression into a numerical expression.
There are $219$ bonus stars in the game. Example

After a weekend of playing his new game, Tearrik has to go to class. During math class, the teacher drew a square on the whiteboard.

The professor asked the class about the difference between the area of the square and its perimeter.

a Help Tearrik to write an algebraic expression for the difference between the area of the square and its perimeter.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["s"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["s^2 - 4s"]}}

b If the side length of a square is $7,$ find the difference between its area and its perimeter.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["21"]}}

b Use the algebraic expression written in Part A. Evaluate the expression when $s=7.$

a Start by recalling the formulas for the area and the perimeter of a square. The area of a square is given by the $square$ of a side. The square that Tearrik's teacher drew indicates that each side has a length of $s.$

$Area of the Square:s_{2} $

The perimeter, on the other hand, is found by adding the length of all sides of a square. Since the four sides of the square all have a length of $s,$ the perimeter is found by multiplying $s$ by $4.$
$Perimeter of the Square:4s $

Then, to find the difference between these two values, subtract the perimeter from the area. $s_{2}−4s $

b The teacher said that the side length of the square is $7.$ This means that $s$ equals $7.$ Then, to find the difference between the area and the perimeter, evaluate the expression for the difference when $s=7.$

$s_{2}−4s$

Substitute

$s=7$

$7_{2}−4⋅7$

CalcPow

Calculate power

$49−4⋅7$

Multiply

Multiply

$49−28$

SubTerms

Subtract terms

$21$

Example

While studying math, Tearrik remembered that he drew a sign to show the prices for the shirts and the pants during his garage sale.

Since Tearrik started studying algebraic expressions, he realized that he could use variables to represent the number of items of clothing he sold. He assigned $s$ for the shirts he sold and $p$ for the pants he sold.

a Help Tearrik write an algebraic expression for the total amount of money he made from selling shirts and pants.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["s","p"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["5s+16p"]}}

b If Tearrik sold $10$ shirts and $4$ pairs of pants, how much did he make?

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"He made","formTextAfter":"dollars.","answer":{"text":["114"]}}

a Determine how much Tearrik would make from each type of clothing. Then, add these amounts to find the total.

b Substitute the appropriate values for the variables in the expression. Then, evaluate.

a Tearrik made money from selling pants and shirts. In other words, the total amount of money he made is the sum of what he got from selling shirts and pants.

$5⋅s $

$16⋅p $

$5s+16p $

b The amount of money that Tearrik made from selling $10$ shirts and $4$ pairs of pants can be found by evaluating the expression.

$Evaluate5s+16p⇓whens=10ands=4 $

Substitute the values for the corresponding variables and find the value of the resulting numerical expression.
$5s+16p$

SubstituteII

$s=10$, $p=4$

$5⋅10+16⋅4$

MultII

Multiply $5$ by $10$

$50+16⋅4$

MultII

Multiply $16$ by $4$

$50+64$

AddTerms

Add terms

$114$

Pop Quiz

Consider the following variables with assigned values.

$a=5c=2 b=21 d=14 $

Evaluate the given algebraic expression for the given values. Closure

At the beginning of the lesson, Tearrik's parents decided to give him money based on what he made from selling his old clothes. How much money Tearrik made initially is unknown, but a variable can be used to represent this value. Consider $x$ as the money Tearrik made at his sale.

$Money From the Garage Sale:x $

Consider what Tearrik's parents said about how much money they would give him. Keep in mind that phrases in this plan will indicate the necessary operations for writing the information as an algebraic expression.
$Doublethe money Tearrik made andaddan additional$5extra. $

The word $double$ indicates multiplication by $2.$ This means that $x$ is multiplied by $2.$
$2x $

On the other hand, the word $add$ indicates addition. The $$5$ are added to double the money Tearrik made.
$2x+5 $

And now an expression for the total amount of money Tearrik got from his garage sale can be written, even if the initial amount is unknown. Algebraic expressions are great for writing mathematical ideas easily.