# Applying Algebraic Properties of Equality  Rating

Ø 4.5 / 2 ratings
The authors Eugene Lee

## Basics on the topicApplying Algebraic Properties of Equality

After this lesson, you will be able to rewrite and modify equations by applying the different properties of equality.

The lesson begins by teaching you how to write an equation representing a real-life problem and to apply the addition property of equality to rewrite it. It leads you to learn how to combine like terms on the same side of the equation and isolate the variable used in the equation by applying the subtraction property of equality. It concludes with determining the solution of the equation by applying the other properties of equality such as the associative properties of multiplication and addition and the commutative property of addition.

Learn about rewriting an equation using the different properties of equality by helping the flobbit Wilbo compute for the length of time that he shall spend on napping so he could complete all his tasks for the day!

This video introduces new concepts, notation, and vocabulary such as the Addition Property (adding equal quantities to both sides of the equation will retain the equality of the two sides); Subtraction Property (subtracting equal quantities from both sides of the equation will retain the equality of the two sides); Associative Property of Multiplication (given a multiplication operation involving more than two factors, the order in which factors are grouped will not affect the product); Associative Property of Addition (given an addition problem involving more than two addends, the order in which addends are grouped will not affect the sum; and Commutative Property of Addition (the order of the addends in an addition problem will not affect the sum).

Before watching this video, you should already be familiar with variables, like terms, equations, and writing an equation to represent a word problem.

After watching this video, you will be prepared to learn how to solve more word problems involving equations such as distance, age, and unknown angle word problems algebraically.

Common Core Standard(s) in focus: 7.EE.B.3 and 7.EE.B.4 A video intended for math students in the 7th grade Recommended for students who are 12-13 years old

## Applying Algebraic Properties of Equality exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Applying Algebraic Properties of Equality.
• ### Explain the addition and subtraction properties of equality.

Hints

To understand the definition of the addition property of equality, if $a=b$ then $a+c=b+c$:

• Let $a=2$, $b=2$ and $c=1$.
• Since $2=2$ then $2+1=2+1$. The equation is still balanced.
To understand the definition of the subtraction property of equality, if $a=b$ then $a-c=b-c$,
• Let $a=2$, $b=2$ and $c=1$.
• Since $2=2$, then $2-1=2-1$. The equation is still balanced.

If we are given the equation $3d+5=10$ and want to add $1$, we add $1$ to both sides to keep it equal. This is called the addition property of equality.

• $3d+5+1=10+1$

If we are given the equation $3d+5=10$ and want to isolate the variable $d$, we first subtract $5$ from both sides. This is called the subtraction property of equality.

• $3d+5-5=10-5$

Solution

• If $a=b$ then $a+c=b+c$.
• This statement means that to keep an equation equal, you have to add the same number to both sides of the equation.
• In Wilbo's equation, $\frac{1}{4}d+210$ = $330$, to add 40 minutes for lunch into the equation, Wilbo needs to add it to the left and right side of the equation to keep the equation equal.
• $\frac{1}{4}d+210+40$ = $330+40$
• Combine like terms: $\frac{1}{4}d+250=370$
• Therefore, Wilbo used the addition property of equality to add lunch into his equation.
2. Subtraction Property of Equality
• If $a=b$ then $a-c=b-c$
• This statement means that to keep an equation equal, you have to subtract the same number from both sides of the equation.
• In Wilbo's equation $\frac{1}{4}d+250=370$, in order to isolate $d$, he needs to subtract 250 from both sides of the equation. This keeps the equation equal.
• $\frac{1}{4}d+250-250=370-250$
• Combine like terms: $\frac{1}{4}d=120$
• Therefore, Wilbo used the subtraction property of equality to isolate the variable $d$.

• ### Match the properties of equalities with their corresponding general equations.

Hints

An example of the Associative Property of Addition is,

• $(2+3)+1=2+(3+1)$
• $6=6$

An example of the Commutative Property of Addition is,

• $2+3=3+2$
• $5=5$

The associative property of multiplication and the commutative property of multiplication are the same as the addition properties but with multiplication instead of addition.

Solution

1. Associative Property of Addition states that the result will be the same regardless of how the numbers are grouped.

• $(a+b)+c=a+(b+c)=b+(a+c)$
Example:
• If $a=1$, $b=2$, $c=3$,
• $(a+b)+c=(1+2)+3=3+3=6$
• $a+(b+c)=1+(2+3)=1+5=6$
• $b+(a+c)=2+(1+3)=2+4=6$
2. Associative Property of Multiplication states that the result will be the same regardless of how the numbers are grouped.
• $(ab)c=a(bc)=b(ac)$
Example:
• If $a=1$, $b=2$, $c=3$,
• $(ab)c=(1 \cdot 2)\cdot 3=2 \cdot 3= 6$
• $a(bc)=1\cdot (2\cdot 3)=1 \cdot 6= 6$
• $b(ac)=2 \cdot (1\cdot 3)=2 \cdot 3= 6$
3. Commutative Property of Addition states that order of addition does not matter.
• $a+b=b+a$
Example:
• If $a=1$ and $b=2$,
• $a+b=1+2=3$
• $b+a=2+1=3$
4. Commutative Property of Multiplication states that the order of multiplication does not matter.
• $ab=ba$
Example:
• If $a=1$ and $b=2$,
• $ab=1\cdot 2=2$
• $ba=2\cdot 1=2$

• ### Determine which statements are true about properties of equalities.

Hints
• Associative properties- the order in which the numbers are grouped does not matter.
• Commutative properties- the order of the numbers does not matter as long as the signs remain the same.
• Equation properties- if you add/subtract a value to one side of the equation, you have to add/subtract the same value to the other side of the equation to keep them equal.

Examples:

• Associative Property of Addition: $(1+2)+3=1+(2+3)$
• Associative Property of Multiplication: $(1\cdot 2)\cdot 3=1\cdot (2\cdot 3)$
• Commutative Property of Addition: $1+2=2+1$
• Commutative Property of Multiplication: $1\cdot 2=2\cdot 1$

Addition Property of Equality: If $a=b$ and you want to add $1$, you have to add it to both sides of the equation, $a+1=b+1$

Subtraction Property of Equality: If $a=b$ and you want to subtract $1$, you have to subtract it to both sides of the equation, $a-1=b-1$.

Solution

True Statements

• $(2+3)+7=2+(3+7)$ is the associative property of addition.
• The addition property of equality states that if $a=b$, then $a+c=b+c$.
• $12\cdot 4=4\cdot 12$ is the commutative property of multiplication.
False Statements

The commutative property of addition states that $a+b=c+a$.

• This statement is false because the commutative property of addition states that $a+b=b+a$.
$7\cdot 4=4\cdot 7$ is the associative property of multiplication.
• This statement is false because $7\cdot 4=4\cdot 7$ is the commutative property of multiplication.
The subtraction property of equality states that if $a=b$, then $a-b=b-a$.
• This statement is false because the subtraction property of equality states that if $a=b$, then $a-c=b-c$.

• ### Label each calculation with the property of equality being used.

Hints
• Associative properties- the order in which the numbers are grouped does not matter.
• Commutative properties- the order of the numbers does not matter as long as the signs remain the same.

Examples:

• Associative Property of Addition: $(15+16)+17=15+(16+17)$
• Associative Property of Multiplication: $(15\cdot 16)\cdot 17=15\cdot (16\cdot 17)$

Examples:

• Commutative Property of Addition: $15+16=16+15$
• Commutative Property of Multiplication: $15\cdot 16=16\cdot 15$

Solution

Associative(+)

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

• $(15+26)+2=15+(26+2)$
• $(54+42)+76=54+(42+76)$
Associative Property of Multiplication

Associative(x)

$(ab)c=a(bc)$

• $(21\cdot 5)\cdot 12=21\cdot(5\cdot 12)$

Commutative(+)

$a+b=b+a$

• $99+17=17+99$
Commutative Property of Multiplication

Commutative(x)

$ab=ba$

• $55\cdot 67=67\cdot 55$
• $100\cdot 68=68\cdot 100$

• ### Identify which expressions equal the given values.

Hints

If we are given the expression, $5+(2+1)$, order of operations tells us to add the parentheses first:

• $5+(2+1)=5+3=8$

Notice that the order in which we add numbers does not change the result.

• $5+2=7$
• $2+5=7$

Multiplication works the same way as addition.

Order of operations tells us to multiply the parentheses first:

• $2\cdot (4\cdot 3)= 2\cdot 12=24$
Notice that the order in which you multiply two numbers does not change the result.
• $3\cdot 2=6$
• $2\cdot 3=6$

Solution

$(5+1)+6=12$ and $5+(1+6)=12$

• Notice that the sum of a group of numbers is the same regardless of how they are grouped. This is called the associative property of addition.
$(4\cdot 2)\cdot 1=8$ and $4\cdot(2\cdot 1)=8$

$4\cdot(3\cdot 1)=12$ and $(4\cdot 3)\cdot 1=12$

• Notice that the product of a group of numbers is the same regardless of how they are grouped. This is called the associative property of multiplication.
$5+3=8$ and $3+5=8$

$9+8=17$ and $8+9=17$

• Notice that the sum of two numbers is the same regardless of what order they are in. This is called the commutative property of addition.
$17\cdot 1=17$ and $1\cdot 17=17$
• Notice that the product of two numbers is the same regardless of what order they are in. The is called the commutative property of multiplication.

• ### Decide which property of equality is being used in each step of the calculation.

Hints

Given: $(3+2x)+4=15$

• $(3+4)+2x=15$ Associative Property of Addition
• $2x+7=15$ Commutative Property of Addition
• $2x+9=17$ Addition Property of Equality

Example of Subtraction Property:

Given: $2x+9=17$

• $2x=8$ Subtraction Property of Equality

Example of Multiplication and Division Properties:

Given: $2x=8$

• $x\cdot 2=8$ Commutative Property of Multiplication
• $x=4$ Division Property of Equality

Solution

Given: $(120+5d)-45=350$

1. $(120-45)+5d=350$

• Associative Property of Addition because different terms on the left side are grouped together.
2. $5d+75=350$
• Commutative Property of Addition because the terms on the left side switch order.
3. $5d+85=360$
• Addition Property of Equality because $10$ is added to both sides to maintain equality.
4. $5d=275$
• Subtraction Property of Equality because $85$ is subtracted from both sides to maintain equality.
5. $d\cdot 5=275$
• Commutative Property of Multiplication because the terms on the left side switch order.
6. $d=55$
• Division Property of Equality because $5$ was divided from both sides of the equation.