Mathematics
Grade 9
15 min
Complete the addition sentence - one digit
Complete the addition sentence - one digit
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Introduction & Learning Objectives
Learning Objectives
Translate a simple arithmetic sentence into a formal algebraic equation with a variable.
Justify each step in solving a linear equation using the fundamental axioms of arithmetic and properties of equality.
Define and apply the concepts of additive identity and additive inverse to isolate a variable.
Prove the uniqueness of the solution for an equation of the form a + x = c where a and c are integers.
Model a one-digit addition problem as a linear function and determine the input that yields a given output.
Distinguish between an axiom, a property, and a procedural shortcut in algebraic manipulation.
You've known that 5 + __ = 9 means the blank is 4 since you were six years old. But can you prove it, mathematically, using the formal rules of logic?...
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Key Concepts & Vocabulary
TermDefinitionExample
Open Sentence (Equation)A mathematical statement containing one or more variables, which can be either true or false depending on the values assigned to the variables.The sentence '3 + x = 8' is an open sentence. It is true if x = 5 and false for any other value of x.
AxiomA statement or proposition which is regarded as being established, accepted, or self-evidently true, serving as a premise or starting point for further reasoning and arguments.The existence of an additive identity (0) is an axiom of the real number system. We accept that a + 0 = a without needing to prove it.
Additive Identity PropertyStates that there exists a unique number, zero (0), such that when it is added to any number 'a', the result is 'a' itself.For the equat...
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Core Formulas
Addition Property of Equality
If \(a = b\), then \(a + c = b + c\) for any \(c\).
This rule allows you to add the same quantity to both sides of an equation without changing its truth. It is the primary tool for isolating a variable.
Associative Property of Addition
\((a + b) + c = a + (b + c)\)
This property states that when adding three or more numbers, the grouping of the numbers does not affect the sum. It is crucial for regrouping terms during equation solving.
Definition of Subtraction
\(a - b = a + (-b)\)
This rule formally defines subtraction as the addition of the additive inverse. It allows us to convert all subtraction problems into addition problems, simplifying the logical structure.
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Challenging
To prove the uniqueness of the solution for 5 + x = 2, one can use a proof by contradiction. The proof begins by assuming two distinct solutions, x₁ and x₂, exist. Which of the following steps correctly uses the axioms to reach a contradiction?
A.If 5 + x₁ = 2 and 5 + x₂ = 2, then 5 + x₁ = 5 + x₂. Adding -5 to both sides yields x₁ = x₂, contradicting that they are distinct.
B.If 5 + x₁ = 2, then x₁ = -3. If 5 + x₂ = 2, then x₂ = -3. Since -3 = -3, there is no contradiction.
C.Assume x₁ ≠ x₂. Then 5 + x₁ ≠ 5 + x₂ by the definition of a well-defined operation. But both equal 2, so 2 ≠ 2, which is a contradiction.
D.Assume x₁ = 2 and x₂ = 5. Then 5 + 2 ≠ 2 and 5 + 5 ≠ 2, so neither is a solution, which is a contradiction.
Challenging
Let '*' be a new binary operation on the set of integers defined as a * b = a + b - 1. What is the additive inverse of the integer 5 under this new operation?
A.-5
B.-4
C.1
D.-3
Challenging
The concept of a 'well-defined operation' is crucial for solving equations. If addition were NOT well-defined, which of the following could be a logical possibility when attempting to solve 4 + x = 7?
A.The equation would have no solution.
B.Adding -4 to '4 + x' might yield a different result than adding -4 to '7', even though 4 + x = 7.
C.The number -4 would no longer be the additive inverse of 4.
D.The associative property would no longer hold true.
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