Question

LOGICAL REASONING Decide whether the statement is true or false If false, rewrite the right-hand side of the equation so the statement is true. $$ -3.5(6.1+8.2) \stackrel{2}{=}-3.5(6.1)-3.5(8.2) $$

Short answer

The statement is true, as both sides of the equation yield the same result of -50.05. Therefore, there's no need to rewrite the right-hand side of the equation.

Step-by-step solution

Identifying and Executing the Operations on the Left Hand Side

Compute the sum in parentheses and multiply it by -3.5: \(-3.5\times(6.1 + 8.2) = -3.5\times14.3 = -50.05\)

Identifying and Executing Operations on the Right Hand Side

Multiple each number in parentheses by -3.5 and compute the sum: \(-3.5\times6.1 + -3.5\times8.2 = -21.35 + -28.7 = -50.05\)

Compare the Results from Both Sides

Compare the resultant values of each side to verify whether they are equal: \(-50.05 = -50.05\)

Key concepts

Distributive Property

The distributive property is a cornerstone of algebra which allows us to multiply a single term by each term within a parenthesis. It acts as a bridge between arithmetic and algebra by distributing the effect of the multiplication over addition or subtraction within the bracket. For instance, if you have an algebraic expression like \( a(b + c) \), you can use the distributive property to expand it to \( ab + ac \).

This property is not only useful in simplification but also in equation solving where you need to eliminate parentheses to isolate and solve for variables. Understanding and applying the distributive property can make dealing with algebraic expressions far less intimidating. Let's look at how it's used in the exercise:

\( -3.5(6.1+8.2) \stackrel{?}{=} -3.5 \times 6.1 + -3.5 \times 8.2 \)

We see that the distributive property is correctly applied on the right side of the equation, multiplying \( -3.5 \) by each term inside the parentheses individually, which simplifies to \( -21.35 - 28.7 \). The streamlined procedure also enhances mental math skills, as students learn to break down more complicated problems into manageable chunks.

Equation Solving

Equation solving is a fundamental skill in algebra involving finding the value(s) of the variable(s) that make the equation true. It involves a series of steps that can handle anything from simple one-step operations to complex multi-variable systems. Essential to mastering equation solving is understanding the properties of equality and the reverse operations that keep the equation balanced.

In the given problem, solving is simplified because both sides of the equation result in a numerical value rather than an algebraic expression needing further manipulation. By computing both sides separately, we confirm if the initial statement holds true:

Left-hand side: \( -3.5 \times (6.1 + 8.2) = -50.05 \)Right-hand side: \( -3.5 \times 6.1 + (-3.5 \times 8.2) = -50.05 \)Both sides equal \(-50.05\), showing the equation is balanced and thus valid. Here, the deductive reasoning processes help reinforce students' problem-solving abilities.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. Unlike equations, expressions don't have an equal sign and they can’t be 'solved' in the same way, but you can simplify them. Algebraic expressions are fundamental as they represent relationships between quantities and provide a way to model real-life situations mathematically.

In our exercise, \( -3.5(6.1+8.2) \) and \( -3.5 \times 6.1 - 3.5 \times 8.2 \) are both examples of algebraic expressions. They showcase how expressions can appear in different formats – factored and expanded – and emphasize the role of the distributive property in transitioning between these forms.

By working through varied algebraic expressions, students enhance their ability to think abstractly and develop the cognitive flexibility to approach and interpret mathematical situations from multiple perspectives. These skills are not only crucial for algebra but also for higher mathematics and problem-solving in general.