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Expressions represent mathematical ideas by using multiple operations, numbers, and variables. However, sometimes there may be more than one way to write the same idea. This lesson teaches how to manipulate an expression to rewrite it different ways and how to determine if several expressions represent the same idea.
### Catch-Up and Review

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

Challenge

Dominika and her friends are playing a tabletop role playing game that they created. There are three different type of coins in the game — copper, silver, and gold.

Each type of coin has a different value. The least valuable coin is the copper one. A gold coin is worth $4$ copper coins and a silver coin is worth $2$ copper coins. Wondering how many copper coins $x$ gold coins and $y$ silver coins are worth, Kriz and Dominika came up with different expressions.$Kriz:Dominika: 4x+2y2(2x+y) $

Who is correct? {"type":"choice","form":{"alts":["Neither","Dominika","Kriz","Both"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}

Discussion

The addition operation has some properties that allow to rewrite an algebraic expression without changing the values when evaluating. The first one has to do with the order of the addition.

Rule

The order in which two or more terms are added does not affect the value of the sum. In other words, the addends can be written in any order.

$a+b=b+a$

$4+5+1✓10 =1+5+4⇓=10✓ $

Since the Commutative Property of Addition is an axiom, it does not need a proof.Discussion

The way three or more terms are grouped when added does not affect the value of the sum.

$(a+b)+c=a+(b+c)$

$(3+9)+412+4✓16 =3+(9+4)⇓=3+13⇓=16✓ $

Since the Associative Property of Addition is an axiom, it does not need a proof.Discussion

Adding $0$ to any number always results in the number itself.

$a+0=a$

Because of this, $0$ is called the Additive Identity.

Consider a number $a.$ By the Reflexive Property of Equality, $a$ is equal to itself.

$a=a $

Let $b$ be another number. If $b$ is added to and subtracted from the left-hand side of the above equation, the equality still holds true.
$a=a⇔a+b−b=a $

Finally, $b−b$ is equal to $0.$
$a+b−b=a⇔a+0=a✓ $

It has been shown that $a+0=a.$ By the Commutative Property of Addition, $0+a$ is also equal to $a.$
Example

Dominika and her friends are playing the tabletop role playing game that they created.

Dominika's character needs to move very far as quickly as possible. It is good that she has upgrades to move faster! Her character can move $50$ feet in one round, but she can roll a die to add additional distance. This additional distance can be written as $d.$$50+d $

In addition to this, Dominika has an upgrade item that adds $15$ feet to the total distance.
$(50+d)+15 $

Help Dominika do the following. a Performing a few steps in the correct order allows to rewrite the expression so that it has two terms. Write the correct order.

{"type":"pair","form":{"alts":[[{"id":0,"text":"First"},{"id":1,"text":"Second"},{"id":2,"text":"Third"}],[{"id":0,"text":"Commutative Property of Addition"},{"id":1,"text":"Associative Property of Addition"},{"id":2,"text":"Evaluate the Addition"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[0,1,2]]}

b Write the expression from Part A.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["d"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["d+65"]}}

a Use the properties of addition to rewrite the expression.

b Write the expression found by following the steps from Part A.

a The given expression can be rewritten by combining the two constant terms. The first step to rewrite it is to arrange the terms so the constant terms are close together. This is done applying the Commutative Property of Addition.

$(50+d)+15⇓(d+50)+15 $

Looking at the expression above, the next step would be to group the constant terms together. This can be done with the Associative Property of Addition.
$(d+50)+15⇓d+(50+15) $

The final step is to complete the addition.
$d+(50+15)⇓d+65 $

Doing this process, the expression has been rewritten as an expression with only two terms. The steps to get to this point can be summarized as follows. Step | Operation |
---|---|

First | Commutative Property of Addition |

Second | Associative Property of Addition |

Third | Evaluate the Addition |

b The resulting expression was written by following the steps from Part A.

$d+65 $

When this expression is evaluated, the result is the same as evaluating the given original expression.
Discussion

Similar to the Properties of Addition, the multiplication operation has some properties that allow to rewrite an algebraic expressions without modifying the results when evaluating.

Rule

The order in which two or more factors are multiplied does not affect the value of the product. That is, the multiplicands can be written in any order.

$a⋅b=b⋅a$

$3⋅2⋅6✓36 =2⋅6⋅3⇓=36✓ $

Since the Commutative Property of Multiplication is an axiom, it does not need a proof.Discussion

The way three or more factors are grouped when multiplied does not affect the value of the product.

$(a⋅b)⋅c=a⋅(b⋅c) $

$(2⋅4)⋅68⋅6✓48 =2⋅(4⋅6)⇓=2⋅24⇓=48✓ $

Since the Associative Property of Multiplication is an axiom, it does not need a proof.Discussion

Any number multiplied by $1$ is equal to the number itself.

$a⋅1=a$

Because of this, the number $1$ is called the Multiplicative Identity.

Consider a number $a.$ By the definition of multiplication, $a$ multiplied by another number $n$ can be written as $n$ times the addition of $a.$

$a⋅n=ntimesa+a+…+a $

If $n=1,$ the sum has only one term.
$a⋅1=1timea $

Therefore, $a⋅1$ is equal to $a.$ Also, by the Commutative Property of Multiplication, $1⋅a=a.$ Discussion

The result of multiplying any number by $0$ is **always** $0.$

$a⋅0=0$

Consider a number $a$ multiplied by zero.

$a⋅0 $

The number zero can be rewritten as the subtraction of any number from itself. For simplicity, the zero can be rewritten as $1−1.$ $a⋅0=a⋅(1−1) $

Then the number $a$ can be distributed to simplify the expression on the right-hand side of the equation.
It has been shown that $a⋅0=0.$ By the Commutative Property of Multiplication, $0⋅a$ is also equal to $0.$
Example

Dominika and Zain are preparing a cooperative attack to take down a strong enemy together.

a Zain's character currently has an effect that diminishes their attack points. If Zain's character has an attack of $z,$ the resulting points can be written as an expression.

$z⋅54 $

Another player gives Zain a boost that multiplies Zain's attack points by $45 .$
$(z⋅54 )⋅45 $

Write the expression for Zain's character's attack points as a single term with a coefficient. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["z"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["z","1z"]}}

b Dominika's character has $d$ attack points. Dominika also has some items that increase attack points. Her total attack points can be written as an expression.

$1.2⋅(d⋅1.5) $

Write this expression as a term with a single coefficient. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["d"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["1.8d"]}}

a Which property would allow the constant terms to be grouped together?

b Which property allows the order of the terms to be changed?

a An expression is given and needs to be rewritten as a single term with a coefficient. The first thing to notice is that there are two multiplications by a constant in the expression. By using the Associative Property of Multiplication, these constant numbers can be grouped together.

$(z⋅54 )⋅45 ⇓z⋅(54 ⋅45 ) $

Now the fractions can be multiplied directly. Notice that the grouped fractions are reciprocals of one another. This means that multiplying them results in $1.$
$z⋅(54 ⋅45 )⇓z⋅1 $

Finally, this expression can be simplified to $z$ by using the Identity Property of Multiplication.
$z⋅1=z $

After the attack modifiers, Zain's character has their original attack points.
b Now it is time to simplify another expression. The constants can be grouped together to multiply them. The order of the variable and the number $1.5$ can be changed using the Commutative Property of Multiplication.

$1.2⋅(d⋅1.5)⇓1.2⋅(1.5⋅d) $

Similar to Part A, the Associative Property of Multiplication can be used to group the constants.
$1.2⋅(1.5⋅d)⇓(1.2⋅1.5)⋅d $

Finally, the product can be found. The expression can be written as a unique term with the variable $d$ and its coefficient.
$(1.2⋅1.5)⋅d=1.8d $

Dominika's character has almost double their original attack power! Example

After a hard battle, Dominika and her friends went to buy some potions for health and magic.

They decided to buy $x$ health potions and $y$ magic potions. Dominika wrote an expression for the total amount of copper coins that they need to buy the potions.$5(2x+y) $

Every health potion has the same price and every magic potion has the same price. a What is the price of a health potion?

{"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":"copper","answer":{"text":["10"]}}

b What is the price of a magic potion?

{"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":"copper","answer":{"text":["5"]}}

a Use the Distributive Property to rewrite the given expression.

b Use the expression found in Part A.

a It is important to understand what the expression represents. The price of $x$ health potions is the product of $x$ and the price of a single health potion.

$Price ofxHealth Potions:price of a health potion⋅x $

Similarly, the price of $y$ magic potions is the product of $y$ and the price of one magic potion.
$Price ofyMagic Potions:price of a magic potion⋅y $

The total price of $x$ health potions and $y$ magic potions is the sum of these products. $price of a health potion⋅x+price of a magic potion⋅y $

However, the given expression is written differently.
$5(2x+y) $

The good thing is that the expression can be rewritten using the Distributive Property. This property shows how the multiplication of a number and a sum can be rewritten as a sum of products.
$a(b+c)=ab+ac $

A similar procedure can be performed on the given expression to rewrite it.
Now that the expression is rewritten, the $x-$term can be compared with the price of $x$ health potions.
$price of a health potion⋅x⇕10x $

As previously stated, the total for the $x$ health potions is the product of $x$ by the price of a single health potion, which is of $10$ copper.
b In Part A, the given expression was rewritten to identify the price of a health potion. The same expression can be used to find the price of a magic potion.

$10x+5y $

This time the focus should be on the term with the variable $y.$ This term can be compared with the price of $y$ magic potions.
$price of a magic potion⋅y⇕5y $

This indicates the total price paid for the $y$ magic potions, which is the result of multiplying $y$ by the price of one magic potion, $5$ copper.
Discussion

In this lesson, different properties of operations were used to rewrite expressions into different expressions that evaluate to the same results. These expressions may be different, but they represent the same quantity and they are called *equivalent expressions*.

Concept

Two or more expressions are equivalent expressions if they have the same result when evaluated. In the case of numerical expressions, two equivalent expressions result in the same number when evaluating the operations.
**every** value of the variables.

$12−62⋅3 =6=6 $

Since both $12−6$ and $2⋅3$ equal $6,$ these expressions are equivalent. In a similar way, two algebraic expressions are equivalent if both expressions result in the same number for $y+42+y+2 $

In both of these expressions, any value for $y$ results in the same number. For example, substitute $5$ for $y.$ $y+4⇓5+4=9 2+y+2⇓2+5+2=9 $

Example

With the loot from their quest, Dominika and her party go to a town to buy some useful equipment. The shop gives prices in copper coins.

Dominika wants to know the total that they need to buy the equipment, but she does not have a calculator nearby. Help Dominika to find the total using mental math.{"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":"copper","answer":{"text":["130"]}}

Rewrite each price as a multiple of $13.$

The sum of the prices can be written as a numerical expression.

$65+39+26 $

When evaluating an expression with mental math, sometimes it is easier to find an equivalent expression with operations that are easier. In this case, the given numbers are all multiples of $13.$ This means that they can be rewritten as the product of some number and $13.$
$13⋅5+13⋅3+13⋅2 $

Next, the Distributive Property can be applied to separate common factor $13$ from each term.
$13(5+3+2) $

This result is an expression with an addition of small numbers and a multiplication. This expression can be solved using the order of operations.
Therefore, Dominika's party needs $130$ copper coins to buy all three items.
Example

Dominika and her party encountered a dragon!

However, instead of a battle, the dragon presented a quiz and asked for equivalent expressions of an algebraic expression.$14(2z+71 ) $

Select all expressions that are equivalent to the expression above. {"type":"multichoice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.40003em;vertical-align:-0.95003em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">7<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.8359999999999999em;vertical-align:-0.686em;\"><\/span><span class=\"mord\">7<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10756em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"mord\">4<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose delimcenter\" style=\"top:0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">8<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">4<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.04398em;\">z<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,3,4]}

Think about the different properties of operations.

Equivalent expressions have the same result for every value of the variables. These expressions are written by applying properties of operations. The properties can be applied to the given expressions to determine which are equivalent to the expression given by the dragon.

This means that this expression

$14(2z+71 )$

Distr

Distribute $14$

$14⋅2z+14⋅71 $

MultII

Multiply $14$ by $2z$

$28z+14⋅71 $

MultII

Multiply $14$ by $71 $

$28z+714 $

ReduceFrac

$ba =b/7a/7 $

$28z+12 $

DivByOne

$1a =a$

$28z+2$

$7(14z +1)$

Distr

Distribute $7$

$7⋅14z +7⋅1$

IdPropMult

Identity Property of Multiplication

$7⋅14z +7$

MultII

Multiply $7$ by $14z $

$147z +7$

ReduceFrac

$ba =b/7a/7 $

$2z +7$

$28z+2 $

This expression is different than the expression $2z+28,$ so these expressions are $28z+2 $

Applying the Commutative Property of Addition to this expression results in the next possible equivalent expression.
$28z+2=2+28z $

The expressions $2+28z $

The constant term is a multiple of $2$ and the coefficient of the variable term is also a multiple of $2.$ Knowing this, both terms can be rewritten.
$2⋅1+2⋅14z $

The $2$ can be separated from both terms by using the Distributive Property.
$2⋅1+2⋅14z=2(1+14z) $

The expression is equivalent to the previous expression, so it is also equivalent to the expression given by the dragon! Closure

At the beginning of this lesson, it was mentioned that in the game there are three different types of coins with different values.

The copper coins are the least valuable. The gold coins are worth $4$ copper coins and the silver coins are worth $2$ copper coins. Wondering how many copper coins $x$ gold coins and $y$ silver coins are worth, Dominika and Kriz proposed two different expressions.$Kriz:Dominika: 4x+2y2(2x+y) $

The total copper coins can be found by adding the copper coins that correspond to $x$ gold coins to the copper coins from the $y$ silver coins. The corresponding copper coins from $x$ gold coins are the result of multiplying the number of gold coins $x$ by the number of copper coins that one gold coin is worth, $4.$
$Gold Coins: 4x $

In a similar way, the number of copper coins equivalent to $y$ silver coins is the product of $2$ and $y.$ $Gold Coins:Silver Coins: 4x2y $

The total copper coins from both types of coins is the sum of these expressions.
$4x+2y $

This means that Kriz is right. This is great, but there is something important to note. Both coefficients of the expression can be rewritten as multiples of $2.$ This factor can then be separated from the addition by using the Distributive Property.
$4x+2y$

Rewrite

Rewrite $4x$ as $2⋅2x$

$2⋅2x+2y$

Rewrite

Rewrite $2y$ as $2⋅1y$

$2⋅2x+2⋅1y$

FactorOut

Factor out $2$

$2(2x+1y)$

IdPropMult

Identity Property of Multiplication

$2(2x+y)$