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Here are a few recommended readings before getting started with this lesson.
Two numerical expressions are equivalent if one of the expressions is obtained from the other by applying properties of addition and multiplication. Knowing this, match each expression in the left column with the equivalent expression.
Apart from the Properties of Equality, the properties of addition and multiplication are also heavily used when rewriting and simplifying expressions or equations. Thus, it is convenient to know these properties. The commutative property of each operation will be discussed first.
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
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
As seen, the commutative property says that the order in which two or more terms are added or multiplied will not change the resulting sum or product. Next, the associative property of addition and multiplication will be presented.
The way three or more terms are grouped when added does not affect the value of the sum.
(a+b)+c=a+(b+c)
The way three or more factors are grouped when multiplied does not affect the value of the product.
Two expressions are equivalent when one of them is obtained by applying properties of addition and multiplication to the other. For each pair of given expressions, determine whether they are equivalent. In the affirmative case, name the property that transforms one expression into the other.
If two real numbers are equal, then one can be substituted for another in any expression.
If a=b, then a can be substituted for b in any expression.
Apply the Substitution Property of Equality to determine who got the correct answer. Start by substituting 77 for F into the conversion formula. Then, substitute the values Tadeo and Magdalena found for C. If a true statement is obtained, the corresponding value is a solution; otherwise, it is not.
C=23
ca⋅b=ca⋅b
Multiply
Calculate quotient
Add terms
C=25
ca⋅b=ca⋅b
Multiply
Calculate quotient
Add terms
In addition to the commutative and associative properties, multiplication has another useful property that helps in the process of simplifying expressions with parentheses. This property is called the Distributive Property.
Multiplying a number by the sum of two or more addends produces the same result as multiplying the number by each addend individually and then adding all the products together.
When solving equations, sometimes more than one step is needed. Both the number of steps and the operations required depend on the complexity of the given equation. For example, consider the following pair of equations.
The general idea is to simplify both sides of the equation and then isolate the variable on one side of the equation. This is usually done by collecting all the variable terms on one side of the equations and combining them. Then, the operations applied to the variable are undone in reverse order.
Distribute 23
2a⋅2=a
Add terms
LHS⋅32=RHS⋅32
Commutative Property of Multiplication
ba⋅ab=1
1⋅a=a
a⋅cb=ca⋅b
Multiply
Calculate quotient
LHS+2y=RHS+2y
Commutative Property of Addition
Add terms
LHS+4=RHS+4
Add terms
LHS/5=RHS/5
Calculate quotient
At Ali's fruit store, strawberries are on sale! Today they are $0.60 cheaper per pound than usual. Magdalena stopped at the store on her way home from school and bought eight pounds of strawberries. She paid a total of $14.40.
Distribute 8
Multiply
LHS+4.8=RHS+4.8
LHS/8=RHS/8
Calculate quotient
After dinner, Magdalena works on a 3D model of a castle using cardboard for a history project. She wants the walls of the ramparts to be 7 inches wide. Additionally, she wants the value of the perimeter of the wall, in inches, to be the same as the value of its area, in square inches.
How high should the wall be?The area of the wall is 7x minus the area of the two small rectangles at the top. Equate the area expression with the perimeter of the wall. Then, solve the equation for x.
Top Length=11
Commutative Property of Addition
Associative Property of Addition
Add terms
LHS−2x=RHS−2x
Commutative Property of Addition
Subtract terms
LHS+4=RHS+4
Add terms
LHS/5=RHS/5
Calculate quotient
Rearrange equation
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