Problem 45 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -9(-t-3) $$

Step-by-Step Solution

Answer

The expression \(-9(-t-3)\), without parentheses, simplifies to \(9t + 27\).

Step 1: Recognize the Distributive Property

The expression given is \(-9(-t - 3)\). In this expression -9 is being multiplied by each term inside the parentheses (-t and -3). This is a case of distribution.

Step 2: Distribute -9 into the Parentheses

Multiply -9 by each term inside the brackets individually. So, \(-9 * -t\) would be \(9t\) and \(-9 * -3\) would become \(27\).

Step 3: Write the Final Answer

After completing the multiplication, the final expression is \(9t + 27\). This has no parentheses, as was required in the problem.

Key Concepts

Multiplication

In mathematics, multiplication is a fundamental operation that involves combining groups of equal sizes. It's a way to add the same number many times. Picture multiplication as the rapid addition of identical numbers.
When we talk about algebraic expressions, multiplication involves combining numerical coefficients with variables. For example, when you multiply \(-9\) by \(-t\), you're expanding or distributing the numerical constant throughout the expression.
This is why multiplication can also help simplify expressions, helping us solve algebraic equations efficiently.

• In our exercise, multiplication converts \(-9(-t)\) into \(9t\), highlighting how combining negatives yields a positive result.
• It simplifies \(-9(-3)\) into \(27\), showing that two negative numbers multiply to form a positive number.

Algebraic Expressions

An algebraic expression is a mathematical phrase combining numbers, variables, and operational symbols. Variables act as placeholders that can represent one or more numbers. For instance, in the expression \(-t\), \(t\) is a variable.
Algebraic expressions are the bedrock of algebra, serving as a means to solve problems where actual numbers may not be known right away. They're implemented through lightweight, flexible statements that express mathematical relationships or conditions.
A deeper understanding of these expressions grants the power to manipulate equations more freely, and distributing terms across an expression reinforces this comprehension.

• In the provided example, \(-9(-t-3)\), each component within the parentheses is distributed separately. This forms new terms through multiplication.
• Understanding and simplifying these expressions make working with variables not only feasible but straightforward.

Negative Numbers

Negative numbers can often seem tricky, but they're just as essential as positive counterparts in mathematics. They represent values less than zero and can be found in various real-world contexts, such as temperatures below freezing or bank overdrafts.
When manipulating algebraic expressions, understanding how negative numbers function is crucial, especially during multiplication or subtraction processes.
In multiplication, two like signs turn into a positive result. And, conversely, multiplying a positive and a negative results in a negative.

• In the example \(-9(-t-3)\): multiplicative operations between two negatives (such as \(-9\) and \(-t\)) yield a positive product, \(9t\).
• Similarly, multiplying \(-9\) by \(-3\) results in \(27\), further highlighting these intrinsic properties of negative numbers.