Problem 35 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ 10(1-3 t) $$

Step-by-Step Solution

Answer

The expression simplifies to \(10 - 30t\).

Step 1: Apply the Distributive Property

The distributive property states that for all real numbers a, b, and c, we have that \( a(b + c) = ab + ac \) and \( a(b - c) = ab - ac \). In this case, we apply the distributive property for \( a = 10, b = 1, c = 3t \), so using the property we can re-write the given expression \( 10(1-3t) \) as \( 10 * 1 - 10 * 3t \).

Step 2: Final Simplification

Now, we simplify the expression to finish the problem. \( 10*1 - 10*3t = 10 - 30t \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are fundamental elements in mathematics, and they are composed of numbers, variables, and operations. In our example, the expression is \(10(1-3t)\). Here, 10 is a coefficient, 1 is a constant, and \(3t\) represents a term with both a coefficient and a variable. Variables, like \(t\) in this expression, are symbols that stand for unknown values or numbers. By combining these elements using arithmetic operations such as addition, subtraction, multiplication, and division, we create algebraic expressions. Understanding these expressions allows us to perform operations, solve equations, and comprehend mathematical relationships. Algebraic expressions form the backbone for further math topics like equations, functions, and analysis. Grasping these basics is crucial for students as they progress in their math education.

Simplification Through the Distributive Property

Simplification is a key process in mathematics that makes expressions easier to work with. When simplifying expressions, one of the powerful techniques we use is the distributive property.

The Distributive Property

The distributive property allows us to multiply a single term by each term inside a parenthesis. This property helps us eliminate parentheses by distributing multiplication over addition or subtraction. For example, in the original expression \(10(1-3t)\), the distributive property lets us compute \(10 \times 1\) and \(10 \times -3t\), leading us to the simplified expression \(10 - 30t\).

Simplifying Algebraic Expressions

Once you have used the distributive property, you continue to simplify by performing any arithmetic calculations, such as multiplication or addition/subtraction of like terms. Simplification results in a clearer, more manageable expression. This process not only helps solve problems more efficiently but also enhances understanding of the underlying mathematical structure.

Importance of Mathematics Education

Mathematics education is not just about learning to compute; it's about developing critical thinking and problem-solving skills.

Building Essential Skills

By studying mathematical concepts like algebraic expressions and the distributive property, students learn to break down complex problems into simpler parts. This skill is invaluable not only in mathematics but in real-life situations where analyzing and solving problems is necessary.

Encouraging Analytical Thinking

Math education encourages students to think logically and systematically. It teaches them to approach problems methodically and recognize patterns and relationships. This analytical thinking is transferrable to other disciplines and everyday life. Incorporating these elements into mathematics education helps in nurturing a generation that can think critically, adapt to new challenges, and approach problems with confidence. Students equipped with these skills tend to have better academic performance and are more prepared for future careers that involve analytics and data interpretation.