Problem 59 - Prealgebra

Question

Write an equation using three integers that is an example of the Distributive Property.

Step-by-Step Solution

Answer

An example of the Distributive Property is: \(2(3 + 4) = 2 \times 3 + 2 \times 4\).

Step 1: Understanding the Distributive Property

The Distributive Property states that for any three numbers, say \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). This property allows us to distribute a multiplication over an addition.

Step 2: Choosing Integers for the Equation

Let's choose three integers for the equation. We'll use \(a = 2\), \(b = 3\), and \(c = 4\). These integers will be substituted into the general form of the distributive property.

Step 3: Formulating the Equation

Substitute the chosen integers into the distributive property form: \(2(3 + 4)\). According to the distributive property, this should equal \(2 \times 3 + 2 \times 4\).

Step 4: Simplifying the Equation

Now, perform the operations to simplify both sides of the equation. Calculate \(2(3 + 4)\), which simplifies to \(2 \times 7 = 14\). On the right side, \(2 \times 3 = 6\) and \(2 \times 4 = 8\). Add these together to get \(6 + 8 = 14\).

Step 5: Verifying the Equality

Both sides of the equation, \(14 = 14\), confirm the use of the distributive property correctly and show that the property holds for these chosen integers.

Key Concepts

Integer Operations

Integer operations are the foundation of working with whole numbers in mathematics. Integers include positive numbers, negative numbers, and zero. Understanding how to perform operations such as addition, subtraction, multiplication, and division with integers is crucial.

• Addition: When adding two positive integers, the result is positive. Adding two negative integers results in a negative value. If you add a positive integer and a negative integer, the result depends on which has the larger absolute value.
• Subtraction: When subtracting integers, it's often easier to think of subtraction as adding the opposite. For example, subtracting a negative is the same as adding a positive.
• Multiplication: Multiplying two integers with the same sign (both positive or both negative) results in a positive integer. If signs differ, the result is negative.
• Division: Division follows similar rules as multiplication regarding sign. Ensure not to divide by zero, as it is undefined.

Mastering these operations is essential for tackling more complex algebraic problems, such as working with the distributive property.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. Variables in these expressions represent unknown values and can be manipulated like numbers.

Consider the algebraic expression \(a(b + c)\). Here, \(a\), \(b\), and \(c\) may be any numbers or variables. The expression illustrates how multiplication can be distributed over addition, showcasing a key property in simplifying equations.

• Variables: Letters representing numbers in an expression (e.g., x, y, a, b).
• Terms: Parts of the expression separated by '+' or '-' signs.
• Coefficients: Numbers multiplied by variables in a term (in \(2x\), '2' is the coefficient).
• Constants: Numbers without variables.

Algebraic expressions require simplification, often using properties like the distributive property, to solve equations efficiently.

Mathematical Equations

Mathematical equations are statements asserting the equality of two expressions, often involving variables and constants. Solving equations involves finding the values of variables that make the equation true. The distributive property is a powerful tool in this process.

In equations such as \(2(3 + 4) = 2 \cdot 3 + 2 \cdot 4\), we see how the distributive property allows multiplication across a sum, simplifying and equating expressions on both sides.

• Simplifying: Combine like terms and use properties of operations to reduce complex expressions.
• Balancing: Maintain equality by performing equivalent operations on both sides of the equation.
• Verification: Always recheck solutions by substituting them back into the original equation.

Grasping the basics of equation construction and solution helps in understanding how mathematical ideas fit into real-world problems.