Question
Why do we have equality properties for addition, subtraction, multiplication and division? How do we know these work for equalities (equations) when we perform an operation to solve for x? The 1st blank of the code is "1" .
make the explanation short
make the explanation short
Answers
Answered by
GPT-4o mini
We have equality properties for addition, subtraction, multiplication, and division because they maintain the balance of an equation. Specifically:
1. **Addition Property of Equality**: If \( a = b \), then \( a + c = b + c \). Adding the same value to both sides keeps them equal.
2. **Subtraction Property of Equality**: If \( a = b \), then \( a - c = b - c \). Subtracting the same value also keeps them equal.
3. **Multiplication Property of Equality**: If \( a = b \), then \( ac = bc \). Multiplying both sides by the same number preserves equality.
4. **Division Property of Equality**: If \( a = b \) and \( c \neq 0 \), then \( \frac{a}{c} = \frac{b}{c} \). Dividing both sides by the same non-zero number maintains equality.
These properties work because they operate symmetrically, ensuring that any operation applied to both sides of an equation doesn't change the truth of the equation. This allows us to manipulate equations confidently to solve for \( x \).
1. **Addition Property of Equality**: If \( a = b \), then \( a + c = b + c \). Adding the same value to both sides keeps them equal.
2. **Subtraction Property of Equality**: If \( a = b \), then \( a - c = b - c \). Subtracting the same value also keeps them equal.
3. **Multiplication Property of Equality**: If \( a = b \), then \( ac = bc \). Multiplying both sides by the same number preserves equality.
4. **Division Property of Equality**: If \( a = b \) and \( c \neq 0 \), then \( \frac{a}{c} = \frac{b}{c} \). Dividing both sides by the same non-zero number maintains equality.
These properties work because they operate symmetrically, ensuring that any operation applied to both sides of an equation doesn't change the truth of the equation. This allows us to manipulate equations confidently to solve for \( x \).