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. That is, the addends can be written in any order.
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.
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 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 then can be substituted for in any expression.
Apply the Substitution Property of Equality to determine who got the correct answer. Start by substituting for into the conversion formula. Then, substitute the values Tadeo and Magdalena found for If a true statement is obtained, the corresponding value is a solution; otherwise, it is not.
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 an equation and then isolate the variable on one side of the equation.
Cross out common factors
Cancel out common factors
Since Equation (II) has no fractions, this step can be omitted.
At Ali's fruit store, strawberries are on sale! Today they are 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
After dinner, Magdalena works on a model of a castle using cardboard for a history project. She wants the walls of the ramparts to be 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.
The area of the wall is 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
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This equation models the situation in which Mark and his friends chose Course got no discount, and paid
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This equation models the situation in which Mark and his friends chose Course got a discount, and paid
Add and subtract terms