Problem 53 - Prealgebra

Question

Perform the indicated operations. $$7(4 x-7)$$

Step-by-Step Solution

Answer

The simplified expression is \(28x - 49\).

Step 1: Distribute the Constant

To solve the expression \(7(4x-7)\), the first step is to distribute the constant 7 to both terms inside the parentheses. Use the distributive property which states \(a(b + c) = ab + ac\). This gives us:\[ 7 \times 4x + 7 \times (-7) \]

Step 2: Multiply Each Term Individually

Multiply the constants and the variable inside the parentheses by 7:\[ 7 \times 4x = 28x \]\[ 7 \times (-7) = -49 \]

Step 3: Combine the Results

Combine the results from Step 2 to rewrite the expression:\[ 28x - 49 \] This expression represents the fully simplified result of performing the distribution from the original exercise.

Key Concepts

Prealgebra

Prealgebra lays the foundation for understanding more advanced mathematical concepts by focusing on basic arithmetic operations and introducing variables. In prealgebra, students encounter problems involving addition, subtraction, multiplication, and division, but with the twist of integrating variables, such as \(x\). This is when math begins to look more like solving puzzles.
To make headway in prealgebra, it’s important to familiarize yourself with basic algebraic properties, like the distributive property. This particular property allows you to multiply a single number by each term within a parenthesis, which simplifies the expression and is a central theme in exercises like this one.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining equivalent value. This process makes mathematical expressions much easier to work with, whether you are solving for a variable or evaluating the expression for a particular value. Here’s how you can tackle the simplification process:

• Apply Distributive Property: Distributing numbers outside parentheses over the terms inside, as shown, is key to breaking down complex expressions.
• Combine Like Terms: Once you use the distributive property, add or subtract the terms to further simplify.

Simplification should never change the value of the original expression. Instead, it should transform it to a form that is easier to evaluate or understand, as done with the expression \(28x - 49\) in the exercise.

Mathematical Operations

Mathematical operations are essential for manipulating numbers and expressions. The main operations—addition, subtraction, multiplication, and division—are building blocks for solving equations. In exercises like our example, multiplying through distributing lets you interactively engage with these operations:

• **Distribute and Multiply:** By distributing the 7 across \(4x-7\), you're multiplying according to the distributive property, which simplifies the equation into numeric or algebraic expressions.
• **Combine Operations:** When you combine operations, you follow specific rules like the order of operations (PEMDAS/BODMAS) to ensure accurate results.

Mastering these operations helps solve and simplify expressions, providing a solid base as you progress to more complex topics in algebra.