Problem 58 - Algebra 1: Concepts and Skills

Question

Determine whether the equation is true or false. $$ (7.8)(1.5)+4.6=16.3 $$

Step-by-Step Solution

Answer

Yes, the equation \(7.8 \times 1.5 + 4.6 = 16.3\) is true.

Step 1: Multiplication

Calculate \(7.8 \times 1.5\) first following the rule of order of operations. This results in 11.7.

Step 2: Addition and Comparison

Add the result, 11.7, to 4.6. This results in a total sum of 16.3. Since the right and the left sides of the equation are equal, the equation is true.

Key Concepts

Order of Operations

Understanding the order of operations is crucial to correctly evaluate mathematical expressions and equations. It’s like a set of traffic rules for navigating the road of numbers and calculations. In mathematics, we use the acronym PEMDAS to represent the sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When looking at the given exercise, the problem \( (7.8)(1.5)+4.6=16.3 \) requires that we follow these rules diligently. So first, we tackle the multiplication within the parentheses: \( 7.8 \times 1.5 \). Afterward, we can proceed with the addition of 4.6 to the product. The diligent application of these steps means we avoid mistakes that could lead to a false equation. Recognizing the correct order of operations allows us to confirm the equation is indeed true.

Arithmetic Operations

Understanding arithmetic operations is fundamental in all areas of mathematics. Arithmetic operations include addition, subtraction, multiplication, and division. They are the building blocks of more complex mathematical concepts and are used across a variety of real-world scenarios.

Let's reflect on the exercise in question. We handle two arithmetic operations here: multiplication and addition. The multiplication \(7.8 \times 1.5\) is executed first, producing 11.7, and then we perform the addition by adding 4.6 to get 16.3. Mastering these operations, and knowing when and how to apply them, leads to the successful evaluation of mathematical statements, such as verifying the truthfulness of equations.

Algebraic Reasoning

Algebraic reasoning involves finding and following patterns in algebra to solve equations and understand relationships among variables. It goes beyond the rote application of arithmetic operations; it's about deciphering the underlying principles and applying logical thinking to work with numerical and variable expressions.

In the context of the exercise, algebraic reasoning isn't just about calculating the numbers; it's reasoning through the equation \( (7.8)(1.5)+4.6=16.3 \) to predict whether the resulting value will uphold the equality. By completing the multiplication and addition steps accurately, we use algebraic reasoning to validate that the original equation stands true. Developing strong algebraic reasoning skills is essential to progress through more advanced mathematical problems where the relationships between numbers are less straightforward.