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Questions and Answers
What is the Property of Equality?
What is the Property of Equality?
- Multiply the same number on each side of the equal sign.
- Subtract the same number from the right side of the equal sign.
- Add, subtract, multiply, or divide the same number on each side of the equal sign. (correct)
- Add the same number to one side of the equation only.
What is the Addition Property of Equality?
What is the Addition Property of Equality?
If a = b, then a + c = b + c.
What is the Subtraction Property of Equality?
What is the Subtraction Property of Equality?
If a = b, then a - c = b - c.
What is the Multiplication Property of Equality?
What is the Multiplication Property of Equality?
What is the Division Property of Equality?
What is the Division Property of Equality?
What is an Inverse Operation?
What is an Inverse Operation?
What is the Inverse Operation of Subtraction?
What is the Inverse Operation of Subtraction?
What is the Inverse Operation of Addition?
What is the Inverse Operation of Addition?
What is the Inverse Operation of Multiplication?
What is the Inverse Operation of Multiplication?
What is the Inverse Operation of Division?
What is the Inverse Operation of Division?
What does the equation x + 6 = -4 represent?
What does the equation x + 6 = -4 represent?
What is the meaning of x + 6 = -4 in terms of the Property of Equality?
What is the meaning of x + 6 = -4 in terms of the Property of Equality?
What does x + 0 = -10 represent?
What does x + 0 = -10 represent?
What does the equation x = -10 signify?
What does the equation x = -10 signify?
What is an equation?
What is an equation?
What does it mean to solve an equation?
What does it mean to solve an equation?
What is a solution in the context of algebra?
What is a solution in the context of algebra?
What are inverse operations?
What are inverse operations?
What does it mean to isolate the variable?
What does it mean to isolate the variable?
What are linear equations in one variable?
What are linear equations in one variable?
How are terms separated in an equation?
How are terms separated in an equation?
What are like terms?
What are like terms?
What is a coefficient?
What is a coefficient?
How can coefficients of like terms be combined?
How can coefficients of like terms be combined?
What is a constant in algebra?
What is a constant in algebra?
What does identity mean in algebra?
What does identity mean in algebra?
What does conditional refer to in algebra?
What does conditional refer to in algebra?
What is a contradiction in algebra?
What is a contradiction in algebra?
What is an equation that is false for every solution?
What is an equation that is false for every solution?
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Study Notes
Properties of Equality
- The Property of Equality allows performing the same operation on both sides of an equation to maintain balance.
- The Addition Property states: If ( a = b ), then ( a + c = b + c ).
- The Subtraction Property states: If ( a = b ), then ( a - c = b - c ).
- The Multiplication Property states: If ( a = b ) and ( c \neq 0 ), then ( a \cdot c = b \cdot c ).
- The Division Property states: If ( a = b ) and ( c \neq 0 ), then ( a \div c = b \div c ).
Inverse Operations
- An inverse operation is the reverse of a given operation, used for solving equations.
- Inverse of subtraction is addition.
- Inverse of addition is subtraction.
- Inverse of multiplication is division.
- Inverse of division is multiplication.
Solving Equations
- Solving an equation involves finding a value (solution) that makes it true.
- An equation is a mathematical statement asserting that two expressions are equal.
- To isolate the variable means getting it alone on one side of the equation for resolution.
Types of Equations
- Linear equations in one variable can be written in the form ( ax + b = c ).
- Terms in an equation are separated by addition and subtraction signs.
- Like terms have the same variable and exponent and can be combined by adding or subtracting.
Essential Definitions
- A coefficient is the numerical factor of a term.
- A constant is a specific number without a variable.
- An identity indicates that an equation is true for all real numbers, such as ( x + 1 = x + 1 ).
- A conditional equation is true for certain values, exemplified by ( 2x - 5 = 9 ) only being true for ( x = 7 ).
- A contradiction arises when an equation has no solutions, represented as an empty set.
Further Concepts
- Inverse operations undo each other, facilitating equation solving.
- The term "solution" refers to the value that satisfies the equation.
- A contradiction signifies that no value can satisfy the given equation, such as ( x + 1 = x + 2 ), which has no solution.
Key Equation Examples
- Given: ( x + 6 = -4 )
- To isolate ( x ), apply properties of equality.
- Result: ( x = -10 ) demonstrates both identity and solving principles.
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