Problem 86 - Elementary Algebra Within Reach

Question

Determine whether the statement is true or false. Justify your answer. Subtracting 0 from each side of an equation yields an equivalent equation.

Step-by-Step Solution

Answer

True. Subtracting 0 from each side of an equation yields an equivalent equation as it applies the additive identity property of 0, which states that for any number a, \(a - 0 = a\).

Step 1: Recognize the properties of zero

The additive identity property of 0 should be used. This property states that for any number a, \(a - 0 = a\). Therefore, subtracting 0 from any number does not change the value of the original number.

Step 2: Apply property to equation

Applying the additive identity property to an equation: If an equation \(a = b\) is given, subtracting 0 from both sides will result in the equation \(a - 0 = b - 0\). By applying the additive identity property, this simplifies to the original equation \(a = b\).

Step 3: Evaluate resulting equation

As the resulting equation \(a - 0 = b - 0\) simplifies back to the original equation \(a = b\), it is clear that subtracting 0 from both sides of an equation yields an equivalent equation. Therefore, the statement is true.

Key Concepts

Additive Identity Property

In algebra, certain properties help us understand how numbers work with each other. One such important property is the **additive identity property** for the number zero. This property tells us that adding or subtracting zero from any number or expression doesn't change its value. Mathematically, we write this as: - For any number or expression, \(a\), \(a + 0 = a\) and \(a - 0 = a\).

This fundamental rule makes calculations easier and shows that zero is a unique number with a special role. It's a sort of 'identity superhero' in math that keeps the value of other numbers intact when it comes into play.

Properties of Zero

The number zero holds several fascinating properties in mathematics. Understanding these properties can help you solve various equations and problems with ease. Let's explore a few key characteristics:

• **Additive Identity**: As mentioned earlier, zero is the additive identity, meaning it can be added to or subtracted from any number without altering that number's value. \(a + 0 = a\) and \(a - 0 = a\).
• **Zero Multiplication**: Another interesting fact is that any number multiplied by zero is zero. Formally, \(a \times 0 = 0\).
• **Division by Zero**: One crucial caution in math is that division by zero is undefined. No number divided by zero is given a value in standard arithmetic.

Zero acts as a neutral point in the number line and plays a critical role in defining how arithmetic operations work.

Algebraic Equations

Algebraic equations are mathematical statements that use variables, numbers, and operations to express relationships between quantities. They are essential tools for solving problems across various fields of mathematics and beyond.

An equation is considered **equivalent** when both sides of the equation represent the same value, ensuring balance. For example, if you start with \(a = b\), you can perform operations like adding or subtracting the same number from both sides without changing the equation's truth value.

• **Maintaining Balance**: This concept of balance means that whatever you do to one side of an equation, you must also do to the other side to maintain equivalence.
• **Solving Techniques**: Common techniques include addition, subtraction, multiplication, or division of terms on both sides to find the unknown variable.

Algebraic equations lay the foundation for understanding many mathematical principles and are essential in problem-solving models.