When solving an equation, we use the properties of equality to isolate the variable on one side of the equation.
Properties of Equality Let [tex]$a, b, c$[/tex] be real numbers.
| Symmetric | If [tex]$a=b$[/tex], then [tex]$b=a$[/tex]. | When the values or expressions in an equation switch sides, they remain equal. |
| Addition | If [tex]$a=b$[/tex], then [tex]$a+c=b+c$[/tex]. | When the same value is added on both sides of an equation, the sums are equal. |
| Subtraction | If [tex]$a=b$[/tex], then [tex]$a-c=b-c$[/tex] | When the same value is subtracted from both sides of an equation, the differences are equal. |
| Multiplication | If [tex]$a=b$[/tex], then [tex]$a c=b c$[/tex] . | When both sides of an equation are multiplied by the same value, the products are equal. |
| Division | If [tex]$a=b$[/tex], then [tex]$\frac{a}{c}=\frac{b}{c}$[/tex]. | When both sides of an equation are divided by the same value, the quotients are equal. |
| Distributive | [tex]$a(b+c)=a b+ac$[/tex] | The product of a value and a sum is equal to the sum of the products of the value and each addend. |
| Zero Product | If [tex]$a c=0$[/tex], then [tex]$a=0$[/tex] or [tex]$c=0$[/tex]. | When a product is equal to zero, at least one of the factors must equal zero.
Answer (2)
The properties of equality are essential rules in mathematics that allow us to manipulate equations while maintaining their equality. Key properties include the symmetric, addition, subtraction, multiplication, division, distributive, and zero product properties, each serving specific purposes in isolating variables or transforming equations. Understanding these properties is crucial for successfully solving equations in various fields such as engineering and economics.
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Symmetric Property: If a = b , then b = a .
Subtraction Property: If a = b , then a − c = b − c .
Multiplication Property: If a = b , then a c = b c .
Distributive Property: a ( b + c ) = ab + a c .
Explanation
Understanding the Problem We are given a table describing the properties of equality. The goal is to fill in the missing parts of the table.
Symmetric Property The symmetric property states: If a = b , then b = a .
Addition Property The addition property states: If a = b , then a + c = b + c .
Subtraction Property The subtraction property states: If a = b , then a − c = b − c .
Multiplication Property The multiplication property states: If a = b , then a c = b c .
Division Property The division property states: If a = b , then c a = c b .
Distributive Property The distributive property states: a ( b + c ) = ab + a c .
Zero Product Property The zero product property states: If a c = 0 , then a = 0 or c = 0 .
Completing the Table Filling in the missing parts of the table:
Symmetric: If a = b , then b = a .
Subtraction: If a = b , then a − c = b − c .
Multiplication: If a = b , then a c = b c .
Distributive: a ( b + c ) = ab + a c .
Examples
Understanding properties of equality is crucial in various fields, such as engineering, physics, and economics, where solving equations is a common task. For example, in structural engineering, you might need to solve equations to determine the forces acting on a bridge. By applying the properties of equality, engineers can manipulate equations to isolate variables and find the values needed to ensure the bridge's stability. Similarly, in economics, these properties are used to solve supply and demand equations to find equilibrium prices and quantities.
