# Applying Algebraic Properties of Equality05:12 minutes

Video Transcript

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Solve Problems Using Expressions, Equations, and Inequalities (3 Videos)

## Applying Algebraic Properties of Equality Exercise

### Would you like to practice what you’ve just learned? Practice problems for this video Applying Algebraic Properties of Equality help you practice and recap your knowledge.

• #### 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.