Mathematics
Grade 9
15 min
Properties of equality
Properties of equality
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1
Introduction & Learning Objectives
Learning Objectives
Identify the Addition, Subtraction, Multiplication, and Division Properties of Equality.
Define the Reflexive, Symmetric, and Transitive Properties of Equality.
Apply the properties of equality to manipulate and solve multi-step linear equations.
Justify each step in solving an equation by naming the property of equality used.
Distinguish between properties of equality and other algebraic properties like the distributive property.
Construct a logical sequence of steps to isolate a variable in an equation with variables on both sides.
Imagine an old-fashioned balancing scale. If you add a 5kg weight to one side, what must you do to the other side to keep it perfectly balanced? ⚖️
The Properties of Equality are the fundamental rules of algebra that ensure...
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Key Concepts & Vocabulary
TermDefinitionExample
EquationA mathematical statement asserting that two expressions are equal. It always contains an equals sign (=).3x + 5 = 14 is an equation.
EqualityThe state where two quantities have the same value. The properties of equality are the rules that preserve this state.The expression 2 + 3 has equality with the expression 5.
Inverse OperationsOperations that 'undo' each other. These are used with the properties of equality to isolate variables.Addition and subtraction are inverse operations. Multiplication and division are inverse operations.
Isolating the VariableThe process of using inverse operations and properties of equality to get a variable by itself on one side of the equation.In the equation x + 4 = 10, we subtract 4 from both sides to isolate x, resu...
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Core Formulas
Addition and Subtraction Properties of Equality
For any real numbers a, b, and c: If a = b, then a + c = b + c and a - c = b - c.
Use these properties to cancel out constant terms or variable terms by adding or subtracting the same value from both sides of the equation. This is the primary method for moving terms across the equals sign.
Multiplication and Division Properties of Equality
For any real numbers a, b, and c (where c ≠ 0): If a = b, then a * c = b * c and a / c = b / c.
Use these properties to eliminate coefficients of variables. If a variable is multiplied by a number, divide both sides by that number. If it's divided, multiply both sides.
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Challenging
To solve the literal equation `m(x - n) = p` for `x`, assuming `m ≠ 0`, what is the most efficient two-step sequence of properties to apply after the given equation?
A.Division Property, then Addition Property
B.Distributive Property, then Subtraction Property
C.Addition Property, then Division Property
D.Symmetric Property, then Addition Property
Challenging
Student 1 solves `5x - 8 = 2x + 7` by first subtracting `2x` from both sides. Student 2 solves the same equation by first adding 8 to both sides. Which statement accurately describes their approaches?
A.Only Student 1's first step is a valid application of a property of equality.
B.Only Student 2's first step is a valid application of a property of equality.
C.Neither student's first step is valid.
D.Both students' first steps are valid applications of properties of equality.
Challenging
An equation is solved: `ax + b = cx + d`. The steps are:
1. `ax - cx + b = d`
2. `x(a - c) + b = d`
3. `x(a - c) = d - b`
4. `x = (d - b) / (a - c)`
Step 2 is justified by the Distributive Property (in reverse, i.e., factoring). For Step 4 to be a valid, defined final step, what condition must be true?
A.d - b > 0
B.a - c = 1
C.x must be an integer
D.a - c ≠ 0
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